Algebra Fundamentals Practice Exam

Question 1 Algebra Fundamentals

What is the product rule for logarithms?

An arithmetic sequence is a sequence where each term differs from the previous one by a constant called the common difference (d). The nth term is aₙ = a₁ + (n − 1)d. A quadratic equation is a polynomial equation of degree 2, written in standard form as ax² + bx + c = 0 where a ≠ 0. log_b(MN) = log_b(M) + log_b(N). The log of a product equals the sum of the logs of the factors. A strict inequality uses < or > (the boundary is not included), while a non-strict inequality uses ≤ or ≥ (the boundary is included).

Question 2 Algebra Fundamentals

Solve: `3x − 4 ≤ 11`

An inverse function f⁻¹ reverses the action of f. If f(a) = b, then f⁻¹(b) = a. Graphically, the inverse is a reflection of f over the line y = x. Add 4: 3x ≤ 15. Divide by 3: x ≤ 5. In interval notation, the solution is (−∞, 5]. Distribute: 2x + 6 = 2x + 6. This simplifies to 0 = 0, which is always true. The equation has infinitely many solutions. The vertical line test states that a graph represents a function if and only if every vertical line intersects the graph at most once. If any vertical line crosses the graph more than once, the relation is not a function.

Question 3 Algebra Fundamentals

What is Cramer's Rule for a 2×2 system?

Function composition combines two functions by using the output of one as the input of the other. It is written as (f ∘ g)(x) = f(g(x)). You first evaluate g(x), then apply f to that result. For the system ax + by = e and cx + dy = f, Cramer's Rule gives x = (ed − bf)/(ad − bc) and y = (af − ce)/(ad − bc), provided the determinant ad − bc ≠ 0. An arithmetic sequence is a sequence where each term differs from the previous one by a constant called the common difference (d). The nth term is aₙ = a₁ + (n − 1)d. log_b(M/N) = log_b(M) − log_b(N). The log of a quotient equals the difference of the logs.

Question 4 Algebra Fundamentals

Solve by substitution: `y = 2x + 1` and `3x + y = 11`

Case 1: 2x − 3 = 7 → x = 5. Case 2: 2x − 3 = −7 → x = −2. The solutions are x = 5 and x = −2. Substitute y: 3x + (2x + 1) = 11. Simplify: 5x + 1 = 11, so x = 2. Then y = 2(2) + 1 = 5. Solution: (2, 5). Factor both the numerator and denominator completely, then cancel any common factors. Always state the restrictions on the variable (values that make the original denominator zero). A logarithm is the inverse of exponentiation. log_b(x) = y means bʸ = x. It answers the question: "To what power must the base b be raised to produce x?"

Question 5 Algebra Fundamentals

What is interval notation?

Multiply every term by the least common denominator (LCD) to clear all fractions, then solve the resulting equation using standard methods. Interval notation is a way to describe a set of numbers using parentheses and brackets. Parentheses ( indicate the endpoint is excluded, and brackets [ indicate the endpoint is included. For example, [2, 5) means all numbers from 2 (included) to 5 (excluded). A rational expression is a fraction in which the numerator and denominator are both polynomials, such as (x² + 1) / (x − 3). It is undefined wherever the denominator equals zero. Multiply by the reciprocal of the divisor: (A/B) ÷ (C/D) = (A/B) · (D/C) = AD / BC. Then simplify by canceling common factors.

Question 6 Algebra Fundamentals

What is a perfect square trinomial?

Rewrite as a compound inequality: −a < x < a. The solution is the open interval (−a, a). Multiply every term by the least common denominator (LCD) to clear all fractions, then solve the resulting equation using standard methods. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². The middle term is always twice the product of the square roots of the first and last terms. An inverse function f⁻¹ reverses the action of f. If f(a) = b, then f⁻¹(b) = a. Graphically, the inverse is a reflection of f over the line y = x.

Question 7 Algebra Fundamentals

Solve for x: `(x/3) + 2 = 5`

Subtract 2 from both sides: x/3 = 3. Multiply both sides by 3: x = 9. An equation has no solution when simplifying leads to a contradiction such as 0 = 5. This means no value of the variable can make the equation true. Distribute first: 8x − 12 + 5x. Then combine like terms: 13x − 12. A logarithm is the inverse of exponentiation. log_b(x) = y means bʸ = x. It answers the question: "To what power must the base b be raised to produce x?"

Question 8 Algebra Fundamentals

What is a piecewise function?

Multiply each term of the first polynomial by every term of the second polynomial (using the distributive property), then combine like terms. For binomials, you can use the FOIL method. Combine like terms — terms with the same variable raised to the same power. Add or subtract their coefficients accordingly. A piecewise function is defined by different expressions for different intervals of the domain. For example, f(x) = x² if x ≥ 0 and f(x) = −x if x < 0. Multiply every term by the least common denominator (LCD) to clear all fractions, then solve the resulting equation using standard methods.

Question 9 Algebra Fundamentals

What is a polynomial?

Multiply each term of the first polynomial by every term of the second polynomial (using the distributive property), then combine like terms. For binomials, you can use the FOIL method. A polynomial is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where all exponents are non-negative integers. For example, 4x³ − 2x + 7. A coefficient is the numerical factor multiplied by a variable in a term. In the term 7x, the coefficient is 7. The Remainder Theorem states that when a polynomial f(x) is divided by (x − c), the remainder equals f(c). If f(c) = 0, then (x − c) is a factor.

Question 10 Algebra Fundamentals

What is the sum or difference of cubes formula?

Sₙ = n(a₁ + aₙ)/2, or equivalently Sₙ = n/2 · [2a₁ + (n − 1)d]. It is the number of terms times the average of the first and last terms. A rational expression is a fraction in which the numerator and denominator are both polynomials, such as (x² + 1) / (x − 3). It is undefined wherever the denominator equals zero. a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). Remember the mnemonic: SOAP (Same, Opposite, Always Positive) for the signs. The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by each term inside a set of parentheses.

Question 11 Algebra Fundamentals

Solve by elimination: `2x + 3y = 12` and `4x − 3y = 6`

When multiplying powers with the same base, add the exponents: aᵐ · aⁿ = aᵐ⁺ⁿ. A variable is a symbol, usually a letter like x or y, that represents an unknown or changeable quantity in a mathematical expression or equation. Add the equations: 6x = 18, so x = 3. Substitute back: 2(3) + 3y = 12, so 3y = 6 and y = 2. Solution: (3, 2). An arithmetic sequence is a sequence where each term differs from the previous one by a constant called the common difference (d). The nth term is aₙ = a₁ + (n − 1)d.

Question 12 Algebra Fundamentals

What is the formula for the sum of the first n terms of a geometric series?

Split into two separate inequalities: x > a or x < −a. The solution in interval notation is (−∞, −a) ∪ (a, ∞). Sₙ = a₁(1 − rⁿ) / (1 − r), provided r ≠ 1. Completing the square is a technique for rewriting a quadratic expression ax² + bx + c in the form a(x − h)² + k. You take half of the coefficient of x, square it, and add and subtract that value. log_b(x) = log_a(x) / log_a(b). This allows you to compute logarithms in any base using a calculator that only has common or natural log keys: log_b(x) = ln(x) / ln(b).

Question 13 Algebra Fundamentals

How do you find the domain of `f(x) = √(x − 2)`?

The expression under the square root must be non-negative: x − 2 ≥ 0, so x ≥ 2. The domain is [2, ∞). An arithmetic sequence is a sequence where each term differs from the previous one by a constant called the common difference (d). The nth term is aₙ = a₁ + (n − 1)d. A rational expression is a fraction in which the numerator and denominator are both polynomials, such as (x² + 1) / (x − 3). It is undefined wherever the denominator equals zero. For the system ax + by = e and cx + dy = f, Cramer's Rule gives x = (ed − bf)/(ad − bc) and y = (af − ce)/(ad − bc), provided the determinant ad − bc ≠ 0.

Question 14 Algebra Fundamentals

What is a system of equations?

An inverse function f⁻¹ reverses the action of f. If f(a) = b, then f⁻¹(b) = a. Graphically, the inverse is a reflection of f over the line y = x. You must reverse the direction of the inequality sign. For example, if −2x > 6, dividing by −2 gives x < −3. A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously. The absolute value |a| represents the distance of a from zero on the number line. It is always non-negative: |a| ≥ 0 for all real numbers a.

Question 15 Algebra Fundamentals

How do you add or subtract polynomials?

Combine like terms — terms with the same variable raised to the same power. Add or subtract their coefficients accordingly. An algebraic expression is a combination of variables, constants, and operations (such as addition, subtraction, multiplication, and division) without an equals sign. For example, 3x + 5 is an algebraic expression. Substitute y: 3x + (2x + 1) = 11. Simplify: 5x + 1 = 11, so x = 2. Then y = 2(2) + 1 = 5. Solution: (2, 5). An inverse function f⁻¹ reverses the action of f. If f(a) = b, then f⁻¹(b) = a. Graphically, the inverse is a reflection of f over the line y = x.

Question 16 Algebra Fundamentals

What is the product rule for exponents?

FOIL stands for First, Outer, Inner, Last. It is a shortcut for multiplying two binomials: (a + b)(c + d) = ac + ad + bc + bd. When multiplying powers with the same base, add the exponents: aᵐ · aⁿ = aᵐ⁺ⁿ. Distribute: 2x + 6 = 2x + 6. This simplifies to 0 = 0, which is always true. The equation has infinitely many solutions. Multiply each term of the first polynomial by every term of the second polynomial (using the distributive property), then combine like terms. For binomials, you can use the FOIL method.

Question 17 Algebra Fundamentals

Factor: `x² + 10x + 25`

The GCF is the largest expression that divides evenly into all terms of a polynomial. Always look for and factor out the GCF as the first step in factoring. This is a perfect square trinomial: (x + 5)², since 5² = 25 and 2 · x · 5 = 10x. A linear equation is an equation in which the highest power of the variable is 1. Its graph is a straight line. The general form is ax + b = 0. The common difference is d = 4 and a₁ = 3. Using aₙ = a₁ + (n−1)d: a₁₀ = 3 + 9(4) = 39.

Question 18 Algebra Fundamentals

Factor: `6x³ + 9x²`

A negative exponent indicates a reciprocal: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8. An algebraic expression is a combination of variables, constants, and operations (such as addition, subtraction, multiplication, and division) without an equals sign. For example, 3x + 5 is an algebraic expression. Rewrite as a compound inequality: −a < x < a. The solution is the open interval (−a, a). The GCF is 3x². Factor it out: 3x²(2x + 3).

Question 19 Algebra Fundamentals

Solve: `|2x − 3| = 7`

Case 1: 2x − 3 = 7 → x = 5. Case 2: 2x − 3 = −7 → x = −2. The solutions are x = 5 and x = −2. The domain is the set of all permissible input values (x-values) for which the function is defined. Common restrictions include division by zero and even roots of negative numbers. log_b(MN) = log_b(M) + log_b(N). The log of a product equals the sum of the logs of the factors. An inconsistent system has no solution. Graphically, this means the lines are parallel and never intersect. The equations represent contradictory conditions.

Question 20 Algebra Fundamentals

What is a constant in an algebraic expression?

Sₙ = n(a₁ + aₙ)/2, or equivalently Sₙ = n/2 · [2a₁ + (n − 1)d]. It is the number of terms times the average of the first and last terms. A constant is a fixed numerical value that does not change. In the expression 2x + 9, the constant is 9. The common difference is d = 4 and a₁ = 3. Using aₙ = a₁ + (n−1)d: a₁₀ = 3 + 9(4) = 39. Any nonzero number raised to the zero power equals 1: a⁰ = 1 (where a ≠ 0). This follows from the quotient rule: aⁿ / aⁿ = aⁿ⁻ⁿ = a⁰ = 1.

Question 21 Algebra Fundamentals

Solve for x: `5(x − 2) = 3x + 4`

Distribute: 5x − 10 = 3x + 4. Subtract 3x: 2x − 10 = 4. Add 10: 2x = 14. Divide by 2: x = 7. First find g(3) = 9. Then f(9) = 2(9) + 1 = 19. So (f ∘ g)(3) = 19. The difference of squares states a² − b² = (a + b)(a − b). It applies when you have two perfect squares separated by subtraction. The range is the set of all possible output values (y-values) that the function can produce.

Question 22 Algebra Fundamentals

Simplify the expression: `4(2x − 3) + 5x`

This is a perfect square trinomial: (x + 5)², since 5² = 25 and 2 · x · 5 = 10x. Distribute first: 8x − 12 + 5x. Then combine like terms: 13x − 12. Split into two cases: x = a or x = −a. If a < 0, the equation has no solution because absolute value cannot be negative. A function is a relation that assigns exactly one output value for every input value. If f(x) = x², each value of x produces exactly one value of f(x).

Question 23 Algebra Fundamentals

How do you find the inverse of a function algebraically?

The natural logarithm has base e (approximately 2.718) and is written as ln(x). It is widely used in calculus and exponential growth/decay models. When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. Combine like terms — terms with the same variable raised to the same power. Add or subtract their coefficients accordingly. Replace f(x) with y, swap x and y, then solve for y. The resulting expression is f⁻¹(x). Always verify by checking that f(f⁻¹(x)) = x.

Question 24 Algebra Fundamentals

What is the power rule for exponents?

A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously. When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. A quadratic equation can have two distinct real solutions, one repeated real solution, or two complex conjugate solutions, depending on the value of the discriminant. This is a difference of squares: x² − 4² = (x + 4)(x − 4).

Question 25 Algebra Fundamentals

What is the sum and product of the roots of `ax² + bx + c = 0`?

When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. log_b(MN) = log_b(M) + log_b(N). The log of a product equals the sum of the logs of the factors. By Vieta's formulas, the sum of the roots is −b/a and the product of the roots is c/a. Using FOIL: First: 2x², Outer: −8x, Inner: 3x, Last: −12. Combine: 2x² − 5x − 12.

Question 26 Algebra Fundamentals

What is the quotient rule for exponents?

A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant called the common ratio (r). The nth term is aₙ = a₁ · rⁿ⁻¹. When dividing powers with the same base, subtract the exponents: aᵐ / aⁿ = aᵐ⁻ⁿ, provided a ≠ 0. A constant is a fixed numerical value that does not change. In the expression 2x + 9, the constant is 9. Here a=2, b=3, c=−2. Discriminant = 9 + 16 = 25. So x = (−3 ± 5)/4, giving x = 1/2 or x = −2.

Question 27 Algebra Fundamentals

What is a function?

Replace f(x) with y, swap x and y, then solve for y. The resulting expression is f⁻¹(x). Always verify by checking that f(f⁻¹(x)) = x. Split into two cases: x = a or x = −a. If a < 0, the equation has no solution because absolute value cannot be negative. An arithmetic sequence is a sequence where each term differs from the previous one by a constant called the common difference (d). The nth term is aₙ = a₁ + (n − 1)d. A function is a relation that assigns exactly one output value for every input value. If f(x) = x², each value of x produces exactly one value of f(x).

Question 28 Algebra Fundamentals

How do you rationalize a denominator containing a single radical?

The difference of squares states a² − b² = (a + b)(a − b). It applies when you have two perfect squares separated by subtraction. Multiply both the numerator and denominator by the radical in the denominator. For example, 1/√3 = (1 · √3)/(√3 · √3) = √3/3. A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously. Substitute y: 3x + (2x + 1) = 11. Simplify: 5x + 1 = 11, so x = 2. Then y = 2(2) + 1 = 5. Solution: (2, 5).

Question 29 Algebra Fundamentals

What is an algebraic expression?

The range is the set of all possible output values (y-values) that the function can produce. Subtract 2 from both sides: x/3 = 3. Multiply both sides by 3: x = 9. An algebraic expression is a combination of variables, constants, and operations (such as addition, subtraction, multiplication, and division) without an equals sign. For example, 3x + 5 is an algebraic expression. A quadratic equation is a polynomial equation of degree 2, written in standard form as ax² + bx + c = 0 where a ≠ 0.

Question 30 Algebra Fundamentals

What is the difference between a strict and non-strict inequality?

A strict inequality uses < or > (the boundary is not included), while a non-strict inequality uses ≤ or ≥ (the boundary is included). The common logarithm is the logarithm with base 10, written as log(x) or log₁₀(x). For example, log(1000) = 3 because 10³ = 1000. An algebraic expression is a combination of variables, constants, and operations (such as addition, subtraction, multiplication, and division) without an equals sign. For example, 3x + 5 is an algebraic expression. Distribute: 2x + 6 = 2x + 6. This simplifies to 0 = 0, which is always true. The equation has infinitely many solutions.

Question 31 Algebra Fundamentals

Solve by factoring: `x² − 5x + 6 = 0`

A dependent system has infinitely many solutions. The equations represent the same line, so every point on that line is a solution. When dividing powers with the same base, subtract the exponents: aᵐ / aⁿ = aᵐ⁻ⁿ, provided a ≠ 0. A strict inequality uses < or > (the boundary is not included), while a non-strict inequality uses ≤ or ≥ (the boundary is included). Factor: (x − 2)(x − 3) = 0. Set each factor to zero: x = 2 or x = 3.

Question 32 Algebra Fundamentals

What is an arithmetic sequence?

The degree is the highest power of the variable in the polynomial. For example, 5x⁴ + 3x² − 1 has degree 4. Split into two separate inequalities: x > a or x < −a. The solution in interval notation is (−∞, −a) ∪ (a, ∞). The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). It gives the solutions to any quadratic equation in the form ax² + bx + c = 0. An arithmetic sequence is a sequence where each term differs from the previous one by a constant called the common difference (d). The nth term is aₙ = a₁ + (n − 1)d.

Question 33 Algebra Fundamentals

What is the vertical line test?

A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². The middle term is always twice the product of the square roots of the first and last terms. The vertical line test states that a graph represents a function if and only if every vertical line intersects the graph at most once. If any vertical line crosses the graph more than once, the relation is not a function. A coefficient is the numerical factor multiplied by a variable in a term. In the term 7x, the coefficient is 7. a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). Remember the mnemonic: SOAP (Same, Opposite, Always Positive) for the signs.

Question 34 Algebra Fundamentals

What is the distributive property?

An inverse function f⁻¹ reverses the action of f. If f(a) = b, then f⁻¹(b) = a. Graphically, the inverse is a reflection of f over the line y = x. A logarithm is the inverse of exponentiation. log_b(x) = y means bʸ = x. It answers the question: "To what power must the base b be raised to produce x?" The common logarithm is the logarithm with base 10, written as log(x) or log₁₀(x). For example, log(1000) = 3 because 10³ = 1000. The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by each term inside a set of parentheses.

Question 35 Algebra Fundamentals

What is the vertex form of a quadratic equation?

Factor both the numerator and denominator completely, then cancel any common factors. Always state the restrictions on the variable (values that make the original denominator zero). The vertex form is y = a(x − h)² + k, where (h, k) is the vertex of the parabola. If a > 0 the parabola opens upward; if a < 0 it opens downward. Rewrite: x² + 6x = −5. Add (6/2)² = 9 to both sides: (x + 3)² = 4. Take square root: x + 3 = ±2, so x = −1 or x = −5. A function is a relation that assigns exactly one output value for every input value. If f(x) = x², each value of x produces exactly one value of f(x).

Question 36 Algebra Fundamentals

What is the greatest common factor (GCF) in factoring?

log_b(x) = log_a(x) / log_a(b). This allows you to compute logarithms in any base using a calculator that only has common or natural log keys: log_b(x) = ln(x) / ln(b). A quadratic equation can have two distinct real solutions, one repeated real solution, or two complex conjugate solutions, depending on the value of the discriminant. FOIL stands for First, Outer, Inner, Last. It is a shortcut for multiplying two binomials: (a + b)(c + d) = ac + ad + bc + bd. The GCF is the largest expression that divides evenly into all terms of a polynomial. Always look for and factor out the GCF as the first step in factoring.

Question 37 Algebra Fundamentals

What is a rational expression?

A rational expression is a fraction in which the numerator and denominator are both polynomials, such as (x² + 1) / (x − 3). It is undefined wherever the denominator equals zero. A piecewise function is defined by different expressions for different intervals of the domain. For example, f(x) = x² if x ≥ 0 and f(x) = −x if x < 0. The degree is the highest power of the variable in the polynomial. For example, 5x⁴ + 3x² − 1 has degree 4. The natural logarithm has base e (approximately 2.718) and is written as ln(x). It is widely used in calculus and exponential growth/decay models.

Question 38 Algebra Fundamentals

What does it mean for an equation to have no solution?

This is a perfect square trinomial: (x + 5)², since 5² = 25 and 2 · x · 5 = 10x. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². The middle term is always twice the product of the square roots of the first and last terms. An equation has no solution when simplifying leads to a contradiction such as 0 = 5. This means no value of the variable can make the equation true. Distribute: 2x + 6 = 2x + 6. This simplifies to 0 = 0, which is always true. The equation has infinitely many solutions.

Question 39 Algebra Fundamentals

How do you factor a trinomial of the form `x² + bx + c`?

The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). It gives the solutions to any quadratic equation in the form ax² + bx + c = 0. Find two numbers that multiply to c and add to b. If those numbers are p and q, then x² + bx + c = (x + p)(x + q). The domain is the set of all permissible input values (x-values) for which the function is defined. Common restrictions include division by zero and even roots of negative numbers. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². The middle term is always twice the product of the square roots of the first and last terms.

Question 40 Algebra Fundamentals

What does it mean when a system of equations is inconsistent?

An inconsistent system has no solution. Graphically, this means the lines are parallel and never intersect. The equations represent contradictory conditions. Factor: (x − 2)(x − 3) = 0. Set each factor to zero: x = 2 or x = 3. Here a=2, b=3, c=−2. Discriminant = 9 + 16 = 25. So x = (−3 ± 5)/4, giving x = 1/2 or x = −2. To evaluate an expression means to substitute specific values for the variables and then simplify using the order of operations to obtain a numerical result.

Question 41 Algebra Fundamentals

What are like terms?

A variable is a symbol, usually a letter like x or y, that represents an unknown or changeable quantity in a mathematical expression or equation. Like terms are terms that have the same variable raised to the same power. For example, 3x² and −5x² are like terms because they both contain x². Like terms can be combined by adding or subtracting their coefficients. Add 4: 3x ≤ 15. Divide by 3: x ≤ 5. In interval notation, the solution is (−∞, 5]. An algebraic expression is a combination of variables, constants, and operations (such as addition, subtraction, multiplication, and division) without an equals sign. For example, 3x + 5 is an algebraic expression.

Question 42 Algebra Fundamentals

What is a coefficient?

Distribute first: 8x − 12 + 5x. Then combine like terms: 13x − 12. Add the equations: 6x = 18, so x = 3. Substitute back: 2(3) + 3y = 12, so 3y = 6 and y = 2. Solution: (3, 2). Combine like terms — terms with the same variable raised to the same power. Add or subtract their coefficients accordingly. A coefficient is the numerical factor multiplied by a variable in a term. In the term 7x, the coefficient is 7.

Question 43 Algebra Fundamentals

What is the formula for the sum of the first n terms of an arithmetic series?

The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by each term inside a set of parentheses. Sₙ = n(a₁ + aₙ)/2, or equivalently Sₙ = n/2 · [2a₁ + (n − 1)d]. It is the number of terms times the average of the first and last terms. First find g(3) = 9. Then f(9) = 2(9) + 1 = 19. So (f ∘ g)(3) = 19. Function composition combines two functions by using the output of one as the input of the other. It is written as (f ∘ g)(x) = f(g(x)). You first evaluate g(x), then apply f to that result.

Question 44 Algebra Fundamentals

Find the 10th term of the arithmetic sequence: 3, 7, 11, 15, …

Sₙ = a₁(1 − rⁿ) / (1 − r), provided r ≠ 1. The common difference is d = 4 and a₁ = 3. Using aₙ = a₁ + (n−1)d: a₁₀ = 3 + 9(4) = 39. log_b(Mⁿ) = n · log_b(M). The log of a power equals the exponent times the log of the base expression. Multiply by the reciprocal of the divisor: (A/B) ÷ (C/D) = (A/B) · (D/C) = AD / BC. Then simplify by canceling common factors.

Question 45 Algebra Fundamentals

What is the quotient rule for logarithms?

A constant is a fixed numerical value that does not change. In the expression 2x + 9, the constant is 9. The Remainder Theorem states that when a polynomial f(x) is divided by (x − c), the remainder equals f(c). If f(c) = 0, then (x − c) is a factor. Subtract 2 from both sides: x/3 = 3. Multiply both sides by 3: x = 9. log_b(M/N) = log_b(M) − log_b(N). The log of a quotient equals the difference of the logs.

Question 46 Algebra Fundamentals

What is the change of base formula?

Add 4: 3x ≤ 15. Divide by 3: x ≤ 5. In interval notation, the solution is (−∞, 5]. log_b(x) = log_a(x) / log_a(b). This allows you to compute logarithms in any base using a calculator that only has common or natural log keys: log_b(x) = ln(x) / ln(b). Sₙ = n(a₁ + aₙ)/2, or equivalently Sₙ = n/2 · [2a₁ + (n − 1)d]. It is the number of terms times the average of the first and last terms. Graph the boundary line (dashed for strict, solid for non-strict). Then test a point not on the line to determine which half-plane to shade.

Question 47 Algebra Fundamentals

How do you multiply two polynomials?

To evaluate an expression means to substitute specific values for the variables and then simplify using the order of operations to obtain a numerical result. Rewrite as a compound inequality: −a < x < a. The solution is the open interval (−a, a). The Remainder Theorem states that when a polynomial f(x) is divided by (x − c), the remainder equals f(c). If f(c) = 0, then (x − c) is a factor. Multiply each term of the first polynomial by every term of the second polynomial (using the distributive property), then combine like terms. For binomials, you can use the FOIL method.

Question 48 Algebra Fundamentals

How do you add or subtract rational expressions with different denominators?

Rewrite as a compound inequality: −a < x < a. The solution is the open interval (−a, a). Find the least common denominator (LCD), rewrite each fraction with the LCD, then add or subtract the numerators. Finally, simplify the resulting expression. A dependent system has infinitely many solutions. The equations represent the same line, so every point on that line is a solution. The difference of squares states a² − b² = (a + b)(a − b). It applies when you have two perfect squares separated by subtraction.

Question 49 Algebra Fundamentals

Solve for x: `3x + 7 = 22`

log_b(x) = log_a(x) / log_a(b). This allows you to compute logarithms in any base using a calculator that only has common or natural log keys: log_b(x) = ln(x) / ln(b). Subtract 7 from both sides: 3x = 15. Divide both sides by 3: x = 5. A piecewise function is defined by different expressions for different intervals of the domain. For example, f(x) = x² if x ≥ 0 and f(x) = −x if x < 0. To evaluate an expression means to substitute specific values for the variables and then simplify using the order of operations to obtain a numerical result.

Question 50 Algebra Fundamentals

If `f(x) = 2x + 1` and `g(x) = x²`, what is `(f ∘ g)(3)`?

An arithmetic sequence is a sequence where each term differs from the previous one by a constant called the common difference (d). The nth term is aₙ = a₁ + (n − 1)d. Multiply both the numerator and denominator by the radical in the denominator. For example, 1/√3 = (1 · √3)/(√3 · √3) = √3/3. First find g(3) = 9. Then f(9) = 2(9) + 1 = 19. So (f ∘ g)(3) = 19. A logarithm is the inverse of exponentiation. log_b(x) = y means bʸ = x. It answers the question: "To what power must the base b be raised to produce x?"