ACT · Math · Preparing for Higher Math: Functions

Nonlinear functions

Nonlinear functions is part of Preparing for Higher Math: Functions in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include composite functions, domain and range restrictions, transformations of a graph and exponential growth or decay model.

Common traps: applying the outer function first; substituting into the wrong variable; multiplying the functions.

Sample questions

Question 1
The number of cells in a culture is 2,400 cells and increases by 5% each year. Which expression gives the number of cells in a culture, in cells, t years from now?
- A.
  2,400(0.05)^t
- B.
  2,400(1 + 5t)
- C.
  2,400(0.95)^t
- D.
  2,400(1.05)^t
Answer: D. 2,400(1.05)^t
A quantity that increases by 5% each year is multiplied by 1 + 0.05 = 1.05 each year. After t years the amount is 2,400(1.05)^t.
Question 2
The point (3, −5) lies on the graph of y = g(x). Which point must lie on the graph of y = −g(x − 1) + 2?
- A.
  (2, 7)
- B.
  (4, −3)
- C.
  (4, 3)
- D.
  (4, 7)
Answer: D. (4, 7)
The x − 1 inside the function shifts the graph right 1, so x = 3 + 1 = 4. The output becomes −(−5) + 2 = 7, giving (4, 7). Choice A shifts left instead of right; B skips the reflection, −5 + 2; C adds 2 before negating, −(−5 + 2).
Question 3
In the standard (x, y) coordinate plane, what is the horizontal asymptote of the graph of f(x) = (6x² − 1)/(3x² + 4)?
- A.
  y = −1/4
- B.
  y = 0
- C.
  y = 2
- D.
  y = 6
Answer: C. y = 2
The numerator and denominator have the same degree, so the horizontal asymptote is the ratio of the leading coefficients: y = 6/3 = 2. Choice A uses the ratio of the constant terms; B applies the rule for a denominator of higher degree; D uses only the numerator's leading coefficient.
Question 4
A rider's height above the ground on a Ferris wheel at a fair is h(t) = 20 − 18cos(πt/5) meters, where t is the time in minutes after the ride starts. How many minutes after the start does the rider first reach the maximum height?
- A.
  5
- B.
  10
- C.
  20
- D.
  38
Answer: A. 5
h is greatest when cos(πt/5) = −1, which first happens when πt/5 = π, so t = 5 minutes, at a height of 38 m. Choice B (10) is the full period, when the rider is back at the bottom; C (20) is the height of the wheel's center, the midline; D (38) is the maximum height, not the time.
Question 5
The function g is defined by g(x) = −x + 4 for x < 1 and g(x) = 2x − 5 for x ≥ 1. For which values of x is g(x) = 3?
- A.
  x = 1 only
- B.
  x = 1 and x = 4
- C.
  No value of x
- D.
  x = 4 only
Answer: D. x = 4 only
Solve each piece and keep only answers inside that piece's interval. −x + 4 = 3 gives x = 1, but that rule needs x < 1, so reject it; in fact g(1) = −3. 2x − 5 = 3 gives x = 4, which satisfies x ≥ 1. Choice A keeps the rejected value; B keeps both; C rejects the valid one too.

Find your real gaps

A few questions show the style. The free scan shows which ACT skills cost you the most points, then builds a plan around them.

Take the free scan