GRE Quiz: Functions Function Notation

20 questions · exam conditions

0:00

Functions Function NotationQuestion 1 of 20

Let $f(x)=2x+3$ and $g(x)=x^2$ . For $x=2$ , what is the value of $f(x)-g(x)$ ?

3

-3

7

1

-1

← Back to quizzes

GRE Quiz

GRE Quiz: Functions Function Notation

Practice Functions Function Notation in GRE 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 Functions Function Notation, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE.

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 $f(x)=2x+3$ and $g(x)=x^2$ . For $x=2$ , what is the value of $f(x)-g(x)$ ?

$3$ (correct answer)
$-3$
$7$
$1$
$-1$

Explanation: This question tests functions and function notation. The functions are f(x) = 2x + 3 and g(x) = x², so f(x) - g(x) means subtract g from f at x=2. Compute f(2) = 4 + 3 = 7, g(2) = 4. Then 7 - 4 = 3. Therefore, the value is 3. A tempting mistake might be to subtract in reverse, 4 - 7 = -3, option B. Another error could be adding them, 7 + 4 = 11, not an option.

Question 2

Two functions are defined by $p(x)=4-x$ and $q(x)=2x+1$ . What is the value of $p(q(3))$ ?

$-7$
$-3$ (correct answer)
$7$
$3$
$-9$

Explanation: This question tests understanding of composite functions and function notation. We have p(x) = 4 - x and q(x) = 2x + 1, and we need to find p(q(3)). First, we find q(3): q(3) = 2(3) + 1 = 6 + 1 = 7. Then we use this result as the input for p: p(7) = 4 - 7 = -3. Therefore, p(q(3)) = -3. A common error would be to compute q(p(3)) instead, which would give a different result, or to make arithmetic errors in the two-step process.

Question 3

A function $g$ is defined by $g(x)=x^2+2x$ . Which of the following equals $g(3y)$ ?

$9y^2+6y$ (correct answer)
$3y^2+2y$
$(3y)^2+2y$
$9y^2+2y$
$3y^2+6y$

Explanation: This question tests functions and function notation. The function g is defined by g(x) = x² + 2x, so to find g(3y), substitute 3y for x. Replace x with 3y: (3y)² + 2*(3y). Compute 9y² + 6y. Therefore, it equals 9y² + 6y. A tempting mistake might be to add coefficients incorrectly, like 3y² + 2y, option B. Another error could be forgetting to square the 3, giving 3y² + 6y, option E.

Question 4

A function $r$ is defined by $r(x)=x^2+1$ . Which of the following equals $r(2k)$ ?

$2k^2+1$
$4k^2+1$ (correct answer)
$(2k)^2$
$k^2+1$
$4k+1$

Explanation: This question tests understanding of functions and function notation with algebraic expressions. The function r is defined by r(x) = x² + 1. To find r(2k), we substitute 2k for x in the expression: r(2k) = (2k)² + 1 = 4k² + 1. Therefore, r(2k) = 4k² + 1. A common error would be to incorrectly square 2k as 2k² instead of 4k², forgetting that (2k)² = 2² × k² = 4k².

Question 5

A function $q$ is defined as follows: $q(n)$ equals the square of $n$ minus 3. What is the value of $q(\tfrac{1}{2})$ ?

$\tfrac{5}{4}$
$-\tfrac{11}{4}$ (correct answer)
$-\tfrac{5}{2}$
$\tfrac{11}{4}$
$-\tfrac{7}{4}$

Explanation: This question tests functions and function notation. The function q is defined by q(n) = n² - 3, meaning square the input and subtract 3. To find q(1/2), replace n with 1/2 in the expression. So, (1/2)² = 1/4, then 1/4 - 3 = 1/4 - 12/4 = -11/4. Therefore, the value is -11/4. A tempting mistake might be to subtract first, like (1/2 - 3)² = (-5/2)² = 25/4, not an option. Another error could be using 1/2 squared as 1/2, giving 1/2 -3 = -5/2, option C.

Question 6

A function $t$ is defined by $t(x)=x^2-4$ . What is the value of $t(a+1)$ in terms of $a$ ?

$a^2-3$
$(a+1)^2-4$
$a^2+2a-3$ (correct answer)
$a^2+1-4$
$a^2+2a+5$

Explanation: This question tests understanding of functions and function notation with variable substitution. The function is defined by t(x) = x² - 4. To find t(a + 1), we substitute (a + 1) for x in the function: t(a + 1) = (a + 1)² - 4. Expanding (a + 1)²: (a + 1)² = a² + 2a + 1. Therefore, t(a + 1) = a² + 2a + 1 - 4 = a² + 2a - 3. A common error would be to expand (a + 1)² incorrectly as a² + 1, forgetting the middle term 2a from the binomial expansion.

Question 7

A function $f$ is defined by $f(x)=3x-5$ . What is the value of $f(4)$ ?

$7$ (correct answer)
$-7$
$17$
$3$
$-17$

Explanation: This question tests understanding of functions and function notation, specifically evaluating a linear function at a given input. The function is defined by f(x) = 3x - 5, which means to find f(x), we multiply the input by 3 and then subtract 5. To find f(4), we substitute x = 4 into the function: f(4) = 3(4) - 5 = 12 - 5 = 7. Therefore, f(4) = 7. A common error would be to confuse the order of operations or to substitute incorrectly, but following the function definition step by step yields the correct answer of 7.

Question 8

Functions $f$ and $g$ are defined by $f(x)=x^2+1$ and $g(x)=2x+3$ . For $x=2$ , what is the value of $f(x)-g(x)$ ?

$0$
$-2$ (correct answer)
$2$
$-4$
$4$

Explanation: This question tests understanding of functions and function notation, specifically evaluating and subtracting two functions at a given point. We have f(x) = x² + 1 and g(x) = 2x + 3, and need to find f(2) - g(2). First, f(2) = 2² + 1 = 4 + 1 = 5. Next, g(2) = 2(2) + 3 = 4 + 3 = 7. Therefore, f(2) - g(2) = 5 - 7 = -2. A common error might be to subtract the functions first and then evaluate, but we need to evaluate each function separately at x = 2 before subtracting.

Question 9

A function $h$ is defined by $h(t)=\dfrac{t-1}{t+3}$ for $t\ne -3$ . What is the value of $h(1)$ ?

$\dfrac{1}{4}$
$0$ (correct answer)
$\dfrac{2}{4}$
$\dfrac{1}{2}$
$\dfrac{2}{3}$

Explanation: This question tests understanding of functions and function notation with a rational expression. The function h is defined by h(t) = (t-1)/(t+3) for t ≠ -3. To find h(1), we substitute 1 for t in the expression: h(1) = (1-1)/(1+3) = 0/4 = 0. Therefore, h(1) = 0. A common error would be to simplify the fraction incorrectly or to confuse the numerator and denominator operations.

Question 10

A function $r$ is defined by $r(x)=|x-3|$ . What is the value of $r(-1)$ ?

$-4$
$2$
$4$ (correct answer)
$-2$
$3$

Explanation: This question tests functions and function notation. The function r is defined by r(x) = |x - 3|, meaning the absolute value of the input minus 3. To find r(-1), replace x with -1 in the expression. So, -1 - 3 = -4, and | -4 | = 4. Therefore, the value is 4. A tempting mistake might be to forget the absolute value, giving -4, option A. Another error could be subtracting in reverse, |3 - (-1)| = |4| = 4, same, but option D -2 might come from partial miscalculation.

Question 11

Two functions are defined by $f(x)=x-5$ and $g(x)=\dfrac{1}{x}$ . What is the value of $g(f(2))$ ?

$-3$
$\dfrac{1}{3}$
$\dfrac{1}{-3}$
$3$
$-\dfrac{1}{3}$ (correct answer)

Explanation: This question tests understanding of composite functions and function notation. We have f(x) = x - 5 and g(x) = 1/x, and we need to find g(f(2)). First, we find f(2): f(2) = 2 - 5 = -3. Then we use this result as the input for g: g(-3) = 1/(-3) = -1/3. Therefore, g(f(2)) = -1/3. A common error would be to compute f(g(2)) instead, or to incorrectly handle the negative sign when taking the reciprocal of -3.

Question 12

Let $f(x)=x+4$ and $g(x)=\dfrac{x}{2}$ . What is the value of $f(g(10))$ ?

$5$
$9$ (correct answer)
$14$
$7$
$12$

Explanation: This question tests functions and function notation, specifically composition. The functions are f(x) = x + 4 and g(x) = x/2, so f(g(10)) means first apply g to 10, then f to that result. Compute g(10) = 10/2 = 5. Then f(5) = 5 + 4 = 9. Therefore, the value is 9. A tempting mistake might be to compose in reverse, g(f(10)) = (10+4)/2 = 7, option D. Another error could be adding before dividing, like (10 + 4)/2 = 7 again.

Question 13

Let $f(x)=x-2$ and $g(x)=3x+1$ . What is the value of $f(g(4))$ ?

$11$ (correct answer)
$13$
$9$
$7$
$3$

Explanation: This question tests functions and function notation, specifically composition. The functions are f(x) = x - 2 and g(x) = 3x + 1, so f(g(4)) means first apply g to 4, then f to that result. Compute g(4) = 3*4 + 1 = 13. Then f(13) = 13 - 2 = 11. Therefore, the value is 11. A tempting mistake might be to compose in reverse, g(f(4)) = g(2) = 7, option D. Another error could be adding instead of subtracting, like 13 + 2 = 15, not an option.

Question 14

A function $f$ is defined by $f(x)=x^2-1$ . Which of the following equals $f(x+1)$ ?

$x^2+1$
$(x+1)^2-1$
$x^2-1+1$
$x^2+2x-1$
$x^2+2x$ (correct answer)

Explanation: This question tests functions and function notation. The function f is defined by f(x) = x² - 1, so to find f(x+1), substitute x+1 for x. Replace x with x+1: (x+1)² - 1. Expand to x² + 2x + 1 - 1 = x² + 2x. Therefore, it equals x² + 2x. A tempting mistake might be to add 1 instead of subtracting, giving x² + 1, option A. Another error could be incomplete expansion, like x² + 2x - 1, option D.

Question 15

Functions $f$ and $g$ are defined by $f(x)=2x+1$ and $g(x)=x-4$ . What is the value of $(f\circ g)(3)$ , where $(f\circ g)(x)=f(g(x))$ ?

$-1$ (correct answer)
$-3$
$3$
$-5$
$5$

Explanation: This question tests understanding of function composition and function notation. We have f(x) = 2x + 1 and g(x) = x - 4, and we need to find (f∘g)(3) = f(g(3)). First, we find g(3): g(3) = 3 - 4 = -1. Then we evaluate f at this result: f(-1) = 2(-1) + 1 = -2 + 1 = -1. Therefore, (f∘g)(3) = -1. A common error would be to compute (g∘f)(3) instead, which would give g(f(3)) = g(7) = 3, but the notation clearly indicates we need f(g(3)).

Question 16

Functions $f$ and $g$ are defined by $f(x)=x-1$ and $g(x)=x^2$ . What is the value of $(g\circ f)(3)$ , where $(g\circ f)(x)=g(f(x))$ ?

$4$ (correct answer)
$3$
$8$
$9$
$2$

Explanation: This question tests understanding of function composition and function notation. The composition (g∘f)(x) means g(f(x)), where we first apply f, then apply g to the result. Given f(x) = x - 1 and g(x) = x², we first find f(3): f(3) = 3 - 1 = 2. Then we apply g to this result: g(f(3)) = g(2) = 2² = 4. Therefore, (g∘f)(3) = 4. A common error would be to compute f(g(3)) instead, which would give f(9) = 8.

Question 17

A function $m$ is defined by $m(x)=|x-5|$ . What is the value of $m(2)$ ?

$-3$
$7$
$3$ (correct answer)
$-7$
$5$

Explanation: This question tests understanding of functions and function notation involving absolute value. The function is defined by m(x) = |x - 5|, which gives the absolute value of the difference between x and 5. To find m(2), we substitute x = 2: m(2) = |2 - 5| = |-3| = 3. Therefore, m(2) = 3. A common error would be to forget the absolute value and answer -3, but absolute value always returns a non-negative result, making the correct answer 3.

Question 18

For all real numbers $x$ , a function $h$ is defined by $h(x)=\dfrac{2x+5}{x-1}$ , where $x\ne 1$ . What is the value of $h(3)$ ?

$\dfrac{11}{2}$ (correct answer)
$\dfrac{2}{11}$
$\dfrac{7}{2}$
$\dfrac{11}{3}$
$\dfrac{5}{2}$

Explanation: This question tests functions and function notation. The function h is defined by h(x) = (2x + 5)/(x - 1), where x ≠ 1, meaning you compute twice the input plus 5 over the input minus 1. To find h(3), replace x with 3 in the expression. So, numerator is 23 + 5 = 11, denominator is 3 - 1 = 2, so 11/2. Therefore, the value is 11/2. A tempting mistake might be to swap numerator and denominator, giving 2/11, which is option B. Another error could be miscalculating numerator as 23 +1=7, giving 7/2, option C.

Question 19

A function $f$ is defined by $f(x)=5-2x$ . Which of the following equals $f(a+1)$ ?

$5-2a+1$
$5-2a-2$
$3-2a$ (correct answer)
$5-\dfrac{2}{a+1}$
$2a-3$

Explanation: This question tests functions and function notation. The function f is defined by f(x) = 5 - 2x, so to find f(a+1), substitute a+1 for x. Replace x with a+1: 5 - 2(a+1). Expand to 5 - 2a - 2 = 3 - 2a. Therefore, it equals 3 - 2a. A tempting mistake might be to substitute incorrectly, like 5 - 2a +1 = 6 - 2a, not an option. Another error could be swapping to 2x - 5, giving 2(a+1) - 5 = 2a - 3, option E.

Question 20

A function $s$ is defined by $s(x)=\dfrac{5x}{2}-4$ . What is the value of $s(6)$ ?

$6$
$\dfrac{11}{2}$
$\dfrac{15}{2}-4=\dfrac{7}{2}$
$11$ (correct answer)
$\dfrac{5}{12}-4$

Explanation: This question tests understanding of functions and function notation with fractions. The function s is defined by s(x) = (5x/2) - 4. To find s(6), we substitute 6 for x: s(6) = (5×6)/2 - 4 = 30/2 - 4 = 15 - 4 = 11. Therefore, s(6) = 11. A common mistake would be to incorrectly handle the order of operations, such as computing 5×(6/2) instead of (5×6)/2, or to make arithmetic errors with the fractions.