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ACT Math Help: Function Definitions And Notation

Review real example questions for Function Definitions And Notation in ACT Math.

Question 1 / 10

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Given the function $f(x) = x^2 - 3x + 4$ , what is the value of $f(-2)$ ?

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Question 1

Given the function $f(x) = x^2 - 3x + 4$ , what is the value of $f(-2)$ ?

2
6
10
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

Given $f(x) = 2x^2 + 3$ , which statement is true about $f(x + 1)$ ?

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

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

A sequence is defined by $a_1 = 3$ and $a_n = 2a_{n-1} - 1$ for $n \geq 2$ . What is $a_5$ ?

17
23
33 (correct answer)
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

A model rocket's height in feet after $t$ seconds is modeled by $h(t) = -16t^2 + 96t$ . What is the height, in feet, 4 seconds after launch?

128 (correct answer)
256
384
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

If $f(x) = 7 - x$ , what is $f(3)$ ?

4 (correct answer)
10
5
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

If $f(x)=x^2-1$ , what is $f(x)+1$ ?

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

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

What is $f(-2)$ if $f(x) = 5x - 2$ ?

-12 (correct answer)
-8
12
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

If $f(x)=\dfrac{x-6}{2}$ , what is the value of $f(0)$ ?

Substitute $x=0$ : $f(0)=\dfrac{0-6}{2}$ .

$3$
$-3$ (correct answer)
$-6$
$0$

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

Given the functions $f(x) = 2x^2 + 1$ and $g(x) = x - 3$ , what is the value of $f(g(4))$ ?

$3$ (correct answer)
$9$
$17$
$33$

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

If $f(x) = 2x^2 - 3x + 4$ , what is $f(0)$ ?

0
1
4 (correct answer)
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.