ACT Math Quiz: Function Definitions And Notation

What this quiz covers

This quiz focuses on Function Definitions And Notation, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

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.

Question 1

1. 2
2. 6
3. 10
4. 14 (correct answer)

Explanation: The correct answer is D (14). Substitute x = −2 into the function: f(−2) = (−2)² − 3(−2) + 4 = 4 + 6 + 4 = 14. The three key steps are: (1) square the input: (−2)² = +4, not −4; (2) multiply: −3(−2) = +6, not −6; (3) add: 4 + 6 + 4 = 14. A (2) comes from treating −3(−2) as −6: 4 − 6 + 4 = 2. B (6) comes from treating (−2)² as −4: −4 + 6 + 4 = 6. C (10) comes from a partial sign error. Pro tip: when substituting a negative value, write every step explicitly — sign errors on squared terms and products are the most common mistakes in function evaluation.

Question 2

1. $f(x + 1) = 2x^2 + 2x + 5$
2. $f(x + 1) = 2x^2 + 2x + 3$
3. $f(x + 1) = 2x^2 + 4x + 5$ (correct answer)
4. $f(x + 1) = 2x^2 + 4x + 3$

Explanation: This tests function notation with a shifted input, where $f(x+1)$ means substitute the entire quantity $x+1$ everywhere $x$ appears in $f(x)=2x^2+3$. That substitution gives $2(x+1)^2+3$, and expanding the square first yields $2(x^2+2x+1)+3$; distributing the 2 across all three terms gives $2x^2+4x+2+3$, which simplifies to $2x^2+4x+5$. The version $2x^2+4x+3$ comes from distributing the 2 to $x^2$ and $2x$ but forgetting that $2\cdot 1=2$ must be added to the 3, and $2x^2+2x+3$ compounds that by also failing to double the middle term. The result $2x^2+2x+5$ shows the same undistributed middle term with the constant handled correctly, which is a partial-distribution error. When a function's input is an expression, wrap it in parentheses before you simplify, expand the power completely, and only then multiply through by the outside coefficient.

Question 3

1. 17
2. 23
3. 33 (correct answer)
4. 65

Explanation: This is a recursive sequences question testing careful step-by-step application of a rule. Choice C (33) is correct — apply the rule aₙ = 2aₙ₋₁ − 1 repeatedly: a₁ = 3, a₂ = 2(3) − 1 = 5, a₃ = 2(5) − 1 = 9, a₄ = 2(9) − 1 = 17, a₅ = 2(17) − 1 = 33. Choice A (17) stops at a₄ — off by one term, likely from miscounting or starting the index at n = 0. Choice B (23) results from an arithmetic error at one step — perhaps computing a₃ = 2(5) + 1 = 11 (adding instead of subtracting): then a₄ = 2(11) − 1 = 21, a₅ = 2(21) − 1 = 41... or some other mid-sequence error. Choice D (65) comes from using aₙ = 2aₙ₋₁ + 1 throughout (adding instead of subtracting): a₂ = 7, a₃ = 15, a₄ = 31, a₅ = 63 ≈ 65. Pro tip: For recursive sequences, build a table of values term by term — there's no shortcut to jumping ahead. Write out each term explicitly: write the term number, then compute carefully. The −1 in this rule is easy to misread as +1, which produces very different values.

Question 4

1. 128 (correct answer)
2. 256
3. 384
4. 512

Explanation: This is a function evaluation question testing substitution into a quadratic height model. Choice A (128) is correct — substitute t = 4: h(4) = −16(4²) + 96(4) = −16(16) + 384 = −256 + 384 = 128 feet. Choice B (256) comes from computing only the negative term: 16 × 16 = 256, or from adding the terms incorrectly: −256 + 384 treated as 256. Choice C (384) comes from evaluating only the positive term 96 × 4 = 384 and ignoring the −16t² term entirely. Choice D (512) likely comes from computing 96 × 4 + 16 × 4² but adding instead of subtracting: 384 + 256 = 640... or from −16(4) + 96(4) = −64 + 384 = 320, then some addition error. Pro tip: Evaluate each term of a polynomial function separately, then combine. The negative sign on the −16t² term is critical — squaring 4 first gives 16, then multiplying by −16 gives −256 (negative, not positive). The rocket is decelerating, so height at t = 4 should be less than at its peak, not more.

Question 5

1. 4 (correct answer)
2. 10
3. 5
4. 1

Explanation: We need to find f(3) when f(x) = 7 - x. To evaluate f(3), we substitute x = 3 into the function: f(3) = 7 - 3. Following order of operations, we calculate 7 - 3 = 4. Therefore, f(3) = 4.

Question 6

1. $x^2$ (correct answer)
2. $(x+1)^2-1$
3. $x^2-2$
4. $x^2+1$

Explanation: We need to find f(x) + 1 when f(x) = x² - 1. Since f(x) = x² - 1, we have f(x) + 1 = (x² - 1) + 1 = x² - 1 + 1 = x². The answer is x².

Question 7

1. -12 (correct answer)
2. -8
3. 12
4. 8

Explanation: We need to find f(-2) when f(x) = 5x - 2. To evaluate f(-2), we substitute x = -2 into the function: f(-2) = 5(-2) - 2. Following order of operations, we calculate 5(-2) = -10, then -10 - 2 = -12. Therefore, f(-2) = -12.

Question 8

1. $3$
2. $-3$ (correct answer)
3. $-6$
4. $0$

Explanation: We need to find f(0) when f(x) = (x - 6)/2. Substituting x = 0 into the function: f(0) = (0 - 6)/2 = -6/2 = -3. The answer is -3. Be careful with the negative sign: 0 - 6 = -6, and -6 divided by 2 is -3.

Question 9

1. $3$ (correct answer)
2. $9$
3. $17$
4. $33$

Explanation: The correct answer is A (3). Evaluate from the inside out: first compute g(4) = 4 − 3 = 1. Then substitute into f: f(1) = 2(1)² + 1 = 2 + 1 = 3. B (9) comes from skipping g and computing f(4) directly: f(4) = 2(16) + 1 = 33... or computing f with x=4 using the wrong formula. C (17) likely comes from computing g(4) = 1 correctly but then applying f incorrectly. D (33) comes from computing f(4) = 2(4²) + 1 = 33, skipping g entirely. For composite functions f(g(x)), always evaluate the inner function first, then substitute that result into the outer function.

Question 10

1. 0
2. 1
3. 4 (correct answer)
4. 2

Explanation: We need to find f(0) when f(x) = 2x² - 3x + 4. To evaluate f(0), we substitute x = 0 everywhere it appears: f(0) = 2(0)² - 3(0) + 4 = 0 - 0 + 4 = 4. The answer is 4.

Question 11

1. $-4$
2. $14$
3. $32$ (correct answer)
4. $41$

Explanation: This is a function evaluation question testing careful substitution into a polynomial. Choice C (32) is correct — substitute x = −3: 2(−3)² − 3(−3) + 5 = 2(9) + 9 + 5 = 18 + 9 + 5 = 32. Two key steps: (−3)² = 9 (squaring a negative gives a positive), and −3(−3) = +9 (negative times negative). Choice A (−4) treats (−3)² as −9 instead of +9: 2(−9) + 9 + 5 = −18 + 9 + 5 = −4. Squaring a negative always gives a positive result. Choice B (14) likely comes from substituting x = 1 or computing the middle term as −3(−3) = −9: 18 − 9 + 5 = 14. Choice D (41) results from computing (−3)² = 9 correctly but then evaluating the middle term as −3(3) = −9 with an extra error in the coefficient, or from other arithmetic mistakes. Pro tip: Always use parentheses when substituting negative values — write f(−3) = 2(−3)² − 3(−3) + 5 with full parentheses. This prevents sign errors, especially around the squared term.

Question 12

1. $-1$
2. $0$ (correct answer)
3. $3$
4. $8$

Explanation: This is a composite functions question testing inside-out evaluation order. Choice B (0) is correct — evaluate the inner function first: g(−2) = 2(−2) + 3 = −4 + 3 = −1. Then evaluate the outer function at that result: f(−1) = (−1)² − 1 = 1 − 1 = 0. Choice A (−1) evaluates only the inner function g(−2) = −1 and stops there, reporting the intermediate result rather than plugging it into f. Choice C (3) evaluates f at the original input instead of at g(−2): f(−2) = (−2)² − 1 = 4 − 1 = 3 — applying f directly to −2 rather than to g(−2). Choice D (8) results from reversing the composition order — computing g(f(−2)): f(−2) = 3, then g(3) = 9, or from a related order-of-operations error. Pro tip: For f(g(x)), always work from the inside out. Evaluate g first, then use that result as the input for f. The notation f(g(−2)) literally means "find g(−2), then find f of that answer."

Question 13

1. $\dfrac{x - 3}{2x + 1}$
2. $\dfrac{3x + 1}{x - 2}$ (correct answer)
3. $\dfrac{x - 2}{3x + 1}$
4. $\dfrac{3x - 1}{x + 2}$

Explanation: This is an inverse functions question testing the swap-and-solve method. Choice B ((3x + 1)/(x − 2)) is correct — to find the inverse, replace f(x) with y and swap x and y: x = (2y + 1)/(y − 3). Solve for y: multiply both sides by (y − 3): x(y − 3) = 2y + 1 → xy − 3x = 2y + 1 → xy − 2y = 3x + 1 → y(x − 2) = 3x + 1 → y = (3x + 1)/(x − 2). Choice A ((x − 3)/(2x + 1)) simply takes the reciprocal of the original function — flipping numerator and denominator — which is NOT the same as finding the inverse. Choice C ((x − 2)/(3x + 1)) is the reciprocal of the correct answer — another flipping error. Choice D ((3x − 1)/(x + 2)) results from sign errors during the algebraic solving step: using −1 instead of +1 and +2 instead of −2 when moving terms. Pro tip: For rational function inverses, always use the swap-and-solve method: swap x and y, then carefully solve for y algebraically. The most common error is simply flipping the fraction — that gives the reciprocal, not the inverse.

Question 14

1. $\dfrac{1}{2}$
2. $\dfrac{7}{2}$ (correct answer)
3. $5$
4. $\dfrac{17}{4}$

Explanation: This is an inverse functions question testing a faster method than algebraically finding f⁻¹(x). Choice B (7/2) is correct — rather than deriving the full inverse function, directly solve f(x) = 5: (3x + 2)/(x − 1) = 5 → 3x + 2 = 5(x − 1) → 3x + 2 = 5x − 5 → 7 = 2x → x = 7/2. So f(7/2) = 5, which means f⁻¹(5) = 7/2. Choice A (1/2) comes from an arithmetic error in the solving step — perhaps writing 3x + 2 = 5x − 5 as 7 = 2x but then computing x = 7/14 = 1/2. Choice C (5) returns the input value — the student confuses f⁻¹(5) with the output of f, or interprets "f⁻¹(5) = 5" from f(5) = something. Choice D (17/4) evaluates f(5) instead of f⁻¹(5): f(5) = (15 + 2)/(5 − 1) = 17/4 — answering the wrong question. Pro tip: The fastest method for evaluating f⁻¹(a) is to solve f(x) = a directly. This avoids the algebraic work of finding the full inverse formula. Cross-multiplying (3x + 2)/(x − 1) = 5 gives 3x + 2 = 5x − 5. Collect x terms: 7 = 2x. Answer: x = 7/2.

Question 15

1. 1
2. 2
3. 4 (correct answer)
4. 5

Explanation: This is a composite functions from a table question testing inside-out evaluation. Choice C (4) is correct — work backward: we need f(g(a)) = 4. First find which input to f gives 4: scanning f(x) values, f(2) = 4. So g(a) must equal 2. Now find which a gives g(a) = 2: scanning g(x) values, g(4) = 2... wait — checking the table (x: 1,2,3,4,5; f: 3,4,1,4,2; wait f(4)=4 too — but we need g(a)=2): g(4) = 2 (confirmed from table). So a = 4. Note: f(2) = 4 AND f(4) = 4, but only g(a) = 2 when a = 4 gives a valid chain. Choice A (1): g(1) = 5, f(5) = 2 ≠ 4. Choice B (2): g(2) = 1, f(1) = 3 ≠ 4. Choice D (5): g(5) = 2, f(2) = 4... wait — this also works! g(5) = 2 (from table: x=5, g=2... let me recheck. The table from the source: x=1:f=3,g=5; x=2:f=4,g=1; x=3:f=1,g=4; x=4:f=5,g=2; x=5:f=2,g=3. So g(4)=2 and f(2)=4 → f(g(4)) = f(2) = 4 ✓. g(5)=3 and f(3)=1 ≠ 4. So only a=4 works. Pro tip: For f(g(a)) questions with tables, always work inside-out: (1) Find what value(s) make f(y) = 4 by scanning the f column. (2) Find which a gives g(a) = that value by scanning the g column. Verify by tracing the full chain.

Question 16

1. left 3 units and down 2 units. (correct answer)
2. right 3 units and down 2 units.
3. left 3 units and up 2 units.
4. right 3 units and up 2 units.

Explanation: This is a graph transformations question testing vertex form of a parabola. Choice A (left 3 units, down 2 units) is correct — for y = f(x + 3) − 2: the "+3" inside the argument shifts the graph opposite to its sign (left 3 units), and the "−2" outside shifts down 2 units. So the vertex moves from (0, 0) to (−3, −2). Choice B (right 3, down 2) gets the vertical direction right but misreads the horizontal: seeing "+3" and thinking "right 3." Choice C (left 3, up 2) gets the horizontal direction right but misreads the vertical sign: seeing "−2" and thinking "up." Choice D (right 3, up 2) gets both directions wrong. Pro tip: Horizontal shifts are counterintuitive — f(x + h) shifts LEFT (not right) by h units. A helpful way to remember: the graph shifts to make the argument equal zero. For f(x + 3), set x + 3 = 0 → x = −3, so the vertex moves to x = −3 (left). Vertical shifts behave naturally: subtracting outside moves down, adding moves up.

Question 17

1. $0$ and $1$ (correct answer)
2. $-1$ and $1$
3. $0$ and $-1$
4. No real solutions

Explanation: The correct answer is A (0 and 1). Compute f(g(x)): f(g(x)) = f(2x + 1) = (2x + 1)² − 4(2x + 1). Expand: (4x² + 4x + 1) − (8x + 4) = 4x² − 4x − 3. Set equal to −3: 4x² − 4x − 3 = −3 → 4x² − 4x = 0 → 4x(x − 1) = 0 → x = 0 or x = 1. B (−1 and 1) results from a sign error in the factoring: 4x(x + 1) = 0 instead of 4x(x − 1) = 0. C (0 and −1) similarly comes from an expansion or sign error. D (no real solutions) results from an arithmetic mistake that produces a negative discriminant. Always distribute fully when expanding a composite function before setting it equal to the target value.

Question 18

1. $-8$
2. $-2$ (correct answer)
3. $4$
4. $2$

Explanation: We need to find f(3) where f(x) = x² - 4x + 1. Substituting x = 3 into the function: f(3) = 3² - 4(3) + 1 = 9 - 12 + 1 = -2. Following the order of operations, we calculate the exponent first, then multiplication, then addition and subtraction from left to right.

Question 19

1. $x^2 + 4x + 1$
2. $x^2 - 8x + 16 + x + 1$
3. $x^2 + x - 16$
4. $(x-4)^2 + (x-4) + 1$ (correct answer)

Explanation: We need to find f(x - 4) when f(x) = x² + x + 1. To find f(x - 4), we replace every x in the original function with (x - 4): f(x - 4) = (x - 4)² + (x - 4) + 1. This expression shows the direct substitution before any expansion. Choice D correctly represents this substitution form. Note that f(x - 4) means substituting (x - 4) for every x, not subtracting 4 from the result.

Question 20

1. $8$ (correct answer)
2. $-22$
3. $22$
4. $-8$

Explanation: We need to find f(5) where f(x) = 3x - 7. To find f(5), we substitute x = 5 into the function: f(5) = 3(5) - 7 = 15 - 7 = 8. The answer is 8.