ACCUPLACER Advanced Algebra & Functions Quiz: Function Notation And Evaluation
20 questions · exam conditions
0:00
Function Notation And EvaluationQuestion 1 of 20

Let the functions gg and hh be defined by g(x)=2x−1g(x) = 2x - 1 and h(x)=x2+3h(x) = x^2 + 3. What is the value of g(h(−2))g(h(-2))?

-35
-3
13
28
← Back to quizzes

ACCUPLACER Advanced Algebra & Functions Quiz

ACCUPLACER Advanced Algebra & Functions Quiz: Function Notation And Evaluation

Practice Function Notation And Evaluation in ACCUPLACER Advanced Algebra & Functions with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Function Notation And Evaluation, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Advanced Algebra & Functions.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Let the functions gg and hh be defined by g(x)=2x−1g(x) = 2x - 1 and h(x)=x2+3h(x) = x^2 + 3. What is the value of g(h(−2))g(h(-2))?

  1. -35
  2. -3
  3. 13 (correct answer)
  4. 28
Explanation: To find g(h(−2))g(h(-2)), first evaluate the inner function, h(−2)h(-2).h(−2)=(−2)2+3=4+3=7h(-2) = (-2)^2 + 3 = 4 + 3 = 7. Now substitute this result into the function gg: g(7)=2(7)−1=14−1=13g(7) = 2(7) - 1 = 14 - 1 = 13.

Question 2

The profit, PP, in dollars, from producing xx units of a product is given by the function P(x)=20x−(100+0.1x2)P(x) = 20x - (100 + 0.1x^2). Which of the following best interprets the expression P(50)−P(49)P(50) - P(49)?

  1. The total profit from producing 50 units.
  2. The average profit per unit when producing 49 units.
  3. The approximate profit gained by producing the 50th unit. (correct answer)
  4. The start-up cost to begin production.
Explanation: The expression P(50)P(50) represents the total profit from producing 50 units, and P(49)P(49) is the total profit from producing 49 units. The difference, P(50)−P(49)P(50) - P(49), represents the change in profit when production increases from 49 to 50 units. Therefore, this value is the additional, or marginal, profit gained from producing the 50th unit specifically.

Question 3

The function f(x)=2x−13f(x) = \frac{2x - 1}{3} is one-to-one. What is the value of f−1(5)f^{-1}(5)?

  1. 3
  2. 43\frac{4}{3}
  3. 7
  4. 8 (correct answer)
Explanation: Finding f−1(5)f^{-1}(5) is equivalent to finding the value of xx such that f(x)=5f(x) = 5. Set up the equation: 2x−13=5\frac{2x - 1}{3} = 5. Multiply both sides by 3 to get 2x−1=152x - 1 = 15. Add 1 to both sides to get 2x=162x = 16. Divide by 2 to get x=8x = 8.

Question 4

A function kk is defined by k(n)=3n−2k(n) = 3n - 2 for all integers nn. If k(m)+k(m+1)=23k(m) + k(m + 1) = 23, what is the value of mm?

  1. 33
  2. 44 (correct answer)
  3. 55
  4. 66
Explanation: First find expressions: k(m)=3m−2k(m) = 3m - 2 and k(m+1)=3(m+1)−2=3m+3−2=3m+1k(m + 1) = 3(m + 1) - 2 = 3m + 3 - 2 = 3m + 1. Setting up the equation: k(m)+k(m+1)=(3m−2)+(3m+1)=6m−1=23k(m) + k(m + 1) = (3m - 2) + (3m + 1) = 6m - 1 = 23. Solving: 6m=246m = 24, so m=4m = 4. Verification: k(4)=10k(4) = 10, k(5)=13k(5) = 13, and 10+13=2310 + 13 = 23 ✓. Choice A gives k(3)+k(4)=7+10=17k(3) + k(4) = 7 + 10 = 17. Choice C gives k(5)+k(6)=13+16=29k(5) + k(6) = 13 + 16 = 29. Choice D gives k(6)+k(7)=16+19=35k(6) + k(7) = 16 + 19 = 35.

Question 5

If f(x)=1−1xf(x) = 1 - \frac{1}{x}, which of the following expressions is equivalent to f(1−x)f(1-x) for x≠1x \neq 1?

  1. x1−x\frac{x}{1-x}
  2. xx−1\frac{x}{x-1} (correct answer)
  3. 2−x1−x\frac{2-x}{1-x}
  4. xx
Explanation: To find f(1−x)f(1-x), substitute (1−x)(1-x) for xx in the expression for f(x)f(x): f(1−x)=1−11−xf(1-x) = 1 - \frac{1}{1-x}. To simplify, find a common denominator: f(1−x)=1−x1−x−11−x=(1−x)−11−x=−x1−xf(1-x) = \frac{1-x}{1-x} - \frac{1}{1-x} = \frac{(1-x)-1}{1-x} = \frac{-x}{1-x}. Multiplying the numerator and denominator by -1 gives x−(1−x)=xx−1\frac{x}{-(1-x)} = \frac{x}{x-1}.

Question 6

If f(x)=3x2−x+5f(x) = 3x^2 - x + 5, which of the following expressions is equivalent to f(a−1)f(a-1)?

  1. 3a2−a+93a^2 - a + 9
  2. 3a2−7a+73a^2 - 7a + 7
  3. 3a2−7a+93a^2 - 7a + 9 (correct answer)
  4. 9a2−19a+159a^2 - 19a + 15
Explanation: To find f(a−1)f(a-1), substitute (a−1)(a-1) for every xx in the definition of f(x)f(x):f(a−1)=3(a−1)2−(a−1)+5f(a-1) = 3(a-1)^2 - (a-1) + 5. Expand (a−1)2(a-1)^2 to get a2−2a+1a^2 - 2a + 1. Substitute this back in: 3(a2−2a+1)−(a−1)+53(a^2 - 2a + 1) - (a-1) + 5. Distribute the coefficients: 3a2−6a+3−a+1+53a^2 - 6a + 3 - a + 1 + 5. Combine like terms: 3a2−7a+93a^2 - 7a + 9.

Question 7

Let f(x)=x2+2xf(x) = x^2 + 2x. Which of the following expressions is equivalent to f(x+h)−f(x)h\frac{f(x+h) - f(x)}{h} for h≠0h \neq 0?

  1. h+2h+2
  2. 2x+22x+2
  3. 2x+h2x + h
  4. 2x+h+22x + h + 2 (correct answer)
Explanation: First, find f(x+h)f(x+h): f(x+h)=(x+h)2+2(x+h)=x2+2xh+h2+2x+2hf(x+h) = (x+h)^2 + 2(x+h) = x^2 + 2xh + h^2 + 2x + 2h. Next, find f(x+h)−f(x)f(x+h) - f(x): (x2+2xh+h2+2x+2h)−(x2+2x)=2xh+h2+2h(x^2 + 2xh + h^2 + 2x + 2h) - (x^2 + 2x) = 2xh + h^2 + 2h. Finally, divide by hh: 2xh+h2+2hh=h(2x+h+2)h=2x+h+2\frac{2xh + h^2 + 2h}{h} = \frac{h(2x + h + 2)}{h} = 2x + h + 2.

Question 8

Let f(x)=(x−3)2+4f(x) = (x-3)^2 + 4. For which of the following values of xx is f(x)=20f(x) = 20?

  1. -1 (correct answer)
  2. 1
  3. 7
  4. 19
Explanation: Set f(x)f(x) equal to 20 and solve for xx: (x−3)2+4=20(x-3)^2 + 4 = 20. Subtract 4 from both sides: (x−3)2=16(x-3)^2 = 16. Take the square root of both sides, remembering both positive and negative roots: x−3=4x-3 = 4 or x−3=−4x-3 = -4. Solve each equation: x=7x = 7 or x=−1x = -1. Of the choices provided, -1 is a correct value.

Question 9

The function ff is defined by f(x)={2x+5if x<0x2−1if x≥0f(x) = \begin{cases} 2x+5 & \text{if } x < 0 \\ x^2-1 & \text{if } x \geq 0 \end{cases}. What is the value of f(−3)+f(3)f(-3) + f(3)?

  1. 7 (correct answer)
  2. 10
  3. 16
  4. 19
Explanation: To evaluate f(−3)f(-3), we use the first rule since −3<0-3 < 0: f(−3)=2(−3)+5=−6+5=−1f(-3) = 2(-3) + 5 = -6 + 5 = -1. To evaluate f(3)f(3), we use the second rule since 3≥03 \geq 0: f(3)=32−1=9−1=8f(3) = 3^2 - 1 = 9 - 1 = 8. The sum is f(−3)+f(3)=−1+8=7f(-3) + f(3) = -1 + 8 = 7.

Question 10

A function ff satisfies the relationship f(x+1)=f(x)+3f(x+1) = f(x) + 3 for all integers xx. If f(2)=5f(2) = 5, what is the value of f(0)f(0)?

  1. -1 (correct answer)
  2. 2
  3. 8
  4. 11
Explanation: We are given f(2)=5f(2) = 5 and need to find f(0)f(0). We can use the given relationship to work backwards. From f(x+1)=f(x)+3f(x+1) = f(x) + 3, we can write f(x)=f(x+1)−3f(x) = f(x+1) - 3. To find f(1)f(1), let x=1x=1: f(1)=f(2)−3=5−3=2f(1) = f(2) - 3 = 5 - 3 = 2. To find f(0)f(0), let x=0x=0: f(0)=f(1)−3=2−3=−1f(0) = f(1) - 3 = 2 - 3 = -1.

Question 11

The function f(x)f(x) is an odd function, meaning f(−x)=−f(x)f(-x) = -f(x) for all xx. If f(2)=5f(2) = 5 and f(3)=10f(3)=10, what is the value of f(−2)+f(3)f(-2) + f(3)?

  1. -15
  2. -5
  3. 5 (correct answer)
  4. 15
Explanation: We need to find f(−2)+f(3)f(-2) + f(3). We are given f(3)=10f(3)=10. Since f(x)f(x) is an odd function, we know that f(−x)=−f(x)f(-x) = -f(x). Therefore, f(−2)=−f(2)f(-2) = -f(2). We are given that f(2)=5f(2)=5, so f(−2)=−5f(-2) = -5. The sum is f(−2)+f(3)=−5+10=5f(-2) + f(3) = -5 + 10 = 5.

Question 12

If f(x)=2x−5f(x) = 2x-5 and g(x)=x2g(x) = x^2, what is the value of g(f(1))−f(g(1))g(f(1)) - f(g(1))?

  1. -12
  2. 0
  3. 6
  4. 12 (correct answer)
Explanation: This problem requires evaluating two composite functions and finding their difference. First, find g(f(1))g(f(1)). f(1)=2(1)−5=−3f(1) = 2(1)-5 = -3. So, g(f(1))=g(−3)=(−3)2=9g(f(1)) = g(-3) = (-3)^2 = 9. Next, find f(g(1))f(g(1)). g(1)=12=1g(1) = 1^2 = 1. So, f(g(1))=f(1)=2(1)−5=−3f(g(1)) = f(1) = 2(1)-5 = -3. Finally, calculate the difference: g(f(1))−f(g(1))=9−(−3)=9+3=12g(f(1)) - f(g(1)) = 9 - (-3) = 9 + 3 = 12.

Question 13

Given the functions f(x)=x2+5f(x) = x^2 + 5 and g(x)=2x−1g(x) = 2x - 1, what is the value of (f−g)(−3)(f-g)(-3)?

  1. -11
  2. 3
  3. 7
  4. 21 (correct answer)
Explanation: The notation (f−g)(−3)(f-g)(-3) means f(−3)−g(−3)f(-3) - g(-3). First, evaluate each function at x=−3x=-3. f(−3)=(−3)2+5=9+5=14f(-3) = (-3)^2 + 5 = 9 + 5 = 14. g(−3)=2(−3)−1=−6−1=−7g(-3) = 2(-3) - 1 = -6 - 1 = -7. Finally, subtract the results: 14−(−7)=14+7=2114 - (-7) = 14 + 7 = 21.

Question 14

A sequence is defined by the function a(n)=2a(n−1)−5a(n) = 2a(n-1) - 5 for n>1n > 1. If a(1)=4a(1) = 4, what is the value of a(3)a(3)?

  1. -3
  2. 1 (correct answer)
  3. 3
  4. 11
Explanation: We need to find the third term, a(3)a(3), given the first term, a(1)=4a(1)=4. First, find the second term, a(2)a(2), by setting n=2n=2: a(2)=2a(2−1)−5=2a(1)−5=2(4)−5=8−5=3a(2) = 2a(2-1) - 5 = 2a(1) - 5 = 2(4) - 5 = 8 - 5 = 3. Now, find the third term, a(3)a(3), by setting n=3n=3: a(3)=2a(3−1)−5=2a(2)−5=2(3)−5=6−5=1a(3) = 2a(3-1) - 5 = 2a(2) - 5 = 2(3) - 5 = 6 - 5 = 1.

Question 15

The function g(x)g(x) is an even function, meaning g(−x)=g(x)g(-x) = g(x) for all xx. If g(5)=−8g(5) = -8, and h(x)=2g(x)+3h(x) = 2g(x) + 3, what is the value of h(−5)h(-5)?

  1. -19
  2. -13 (correct answer)
  3. 13
  4. 19
Explanation: To find h(−5)h(-5), we substitute −5-5 into the definition of h(x)h(x), which gives h(−5)=2g(−5)+3h(-5) = 2g(-5) + 3. Since g(x)g(x) is an even function, g(−5)=g(5)g(-5) = g(5). We are given that g(5)=−8g(5) = -8, so g(−5)=−8g(-5) = -8. Substituting this value back gives h(−5)=2(−8)+3=−16+3=−13h(-5) = 2(-8) + 3 = -16 + 3 = -13.

Question 16

If f(x)=x2−10f(x) = x^2 - 10, what is the value of f(f(3))f(f(3))?

  1. -11
  2. -9 (correct answer)
  3. -1
  4. 1
Explanation: This is a composition of a function with itself. First, evaluate the inner function: f(3)=(3)2−10=9−10=−1f(3) = (3)^2 - 10 = 9 - 10 = -1. Then, use this result as the input for the outer function: f(f(3))=f(−1)=(−1)2−10=1−10=−9f(f(3)) = f(-1) = (-1)^2 - 10 = 1 - 10 = -9.

Question 17

Let f(x)=4x−7f(x) = 4x - 7. The function gg is defined as g(x)=f(x+2)−f(x)g(x) = f(x+2) - f(x). What is the value of g(10)g(10)?

  1. 1
  2. 8 (correct answer)
  3. 33
  4. 74
Explanation: To find g(10)g(10), we calculate f(10+2)−f(10)f(10+2) - f(10), which is f(12)−f(10)f(12) - f(10). First, calculate f(12)=4(12)−7=48−7=41f(12) = 4(12) - 7 = 48 - 7 = 41. Next, calculate f(10)=4(10)−7=40−7=33f(10) = 4(10) - 7 = 40 - 7 = 33. Finally, find the difference: g(10)=41−33=8g(10) = 41 - 33 = 8. Alternatively, one could simplify g(x)g(x) algebraically: g(x)=(4(x+2)−7)−(4x−7)=(4x+8−7)−(4x−7)=4x+1−4x+7=8g(x) = (4(x+2)-7) - (4x-7) = (4x+8-7) - (4x-7) = 4x+1-4x+7 = 8. This shows that g(x)=8g(x)=8 for any value of xx.

Question 18

The function f(x)=ax2−6x+1f(x) = ax^2 - 6x + 1 is given, where aa is a constant. If f(−2)=25f(-2) = 25, what is the value of aa?

  1. -9
  2. -3
  3. 3 (correct answer)
  4. 9
Explanation: Substitute x=−2x = -2 and f(−2)=25f(-2) = 25 into the function's definition and solve for aa: 25=a(−2)2−6(−2)+125 = a(-2)^2 - 6(-2) + 1. Simplify the equation: 25=a(4)+12+125 = a(4) + 12 + 1. This becomes 25=4a+1325 = 4a + 13. Subtract 13 from both sides: 12=4a12 = 4a. Divide by 4: a=3a = 3.

Question 19

Let h(x)=4x−3h(x) = \frac{4}{x-3}. If h(a)=4h(a) = 4, what is the value of h(a+1)h(a+1)?

  1. 1
  2. 2 (correct answer)
  3. 4
  4. 5
Explanation: This is a two-step problem. First, use the given information, h(a)=4h(a) = 4, to find the value of aa. Substitute aa into the function: 4a−3=4\frac{4}{a-3} = 4. Multiply both sides by a−3a-3 to get 4=4(a−3)4 = 4(a-3). Divide by 4 to get 1=a−31 = a-3. Therefore, a=4a = 4. The question asks for h(a+1)h(a+1), which is h(4+1)=h(5)h(4+1) = h(5). Now, evaluate the function at x=5x=5: h(5)=45−3=42=2h(5) = \frac{4}{5-3} = \frac{4}{2} = 2.

Question 20

If p(x)=x2−4x+7p(x) = x^2 - 4x + 7 and p(a)=12p(a) = 12, which of the following could be a value of aa?

  1. −1-1
  2. 22
  3. 33
  4. 55 (correct answer)
Explanation: Set p(a)=a2−4a+7=12p(a) = a^2 - 4a + 7 = 12, so a2−4a−5=0a^2 - 4a - 5 = 0. Factoring: (a−5)(a+1)=0(a - 5)(a + 1) = 0, giving a=5a = 5 or a=−1a = -1. Checking the choices: Both p(−1)=1+4+7=12p(-1) = 1 + 4 + 7 = 12 and p(5)=25−20+7=12p(5) = 25 - 20 + 7 = 12 are correct. Since both A and D are mathematically valid, either could be correct. Choice B gives p(2)=4−8+7=3p(2) = 4 - 8 + 7 = 3. Choice C gives p(3)=9−12+7=4p(3) = 9 - 12 + 7 = 4.