Inverse Functions Worksheet with Answers – Precalculus (Free PDF)

Problem 1 Evaluate with Function Notation ★★★

Let f(x) = 2x² − 5x + 7. Find f(−3) and f(4).

Problem 2 Domain Using Two Restrictions ★★★

Write the domain in interval notation. (a) g(x) = √(4x + 9)   (b) h(x) = 1x + 6

Problem 3 Domain and Range from a Graph ★★★

The graph shows a function with a closed dot at (−3, 0), a maximum at (1, 4), and an open dot at (5, 0). State the domain and the range in interval notation.

Problem 4 Evaluate Two Compositions ★★★

Let f(x) = x² − 3 and g(x) = 4x + 1. Find f(g(2)) and g(f(2)).

Problem 5 Build Composite Functions ★★★

Let f(x) = 6x + 1 and g(x) = x² − 4. Write (f ∘ g)(x) and (g ∘ f)(x) in simplest form.

Problem 6 Inverse of a Linear Function ★★★

Find the inverse of f(x) = 54x + 3.

Problem 7 Reading Inverse Values from a Table ★★★

A one-to-one function f has this table.

Find f⁻¹(15), f⁻¹(30) and f(f⁻¹(22)). Then explain whether f⁻¹(10) can be found from the table.

Problem 8 Inverse Values from a Graph ★★★

The graph of f passes through (−1, −4), (1, 0) and (3, 4). Use it to find f⁻¹(4), f⁻¹(0) and f⁻¹(−4).

Problem 9 Domain of a Root over a Binomial ★★★

Find the domain of k(x) = √(x + 5)x − 2 in interval notation.

Problem 10 Verify Inverse Functions ★★★

Show that f(x) = 2x³ + 1 and g(x) = ∛(x − 12) are inverses of each other.

Problem 11 Decompose a Composite Function ★★★

Write each function as f(g(x)) with a simple inner and outer function. (a) h(x) = (5x − 2)⁴   (b) k(x) = 1x² + 9

Problem 12 Restrict, Then Invert ★★★

The graph of f(x) = (x + 3)² − 4 is shown. Restrict the domain to x ≥ −3 and find the inverse, with its domain and range.

Problem 13 Inverse of a Rational Function ★★★

Find the inverse of f(x) = x + 1x − 4 and state which input is excluded from the domain of f⁻¹.

Problem 14 Domain of a Composite ★★★

Let f(x) = √(x − 1) and g(x) = x² − 10. Find (f ∘ g)(x) and its domain.

Problem 15 Horizontal Line Test and Restriction ★★★

The graph of g(x) = x³ − 3x is shown with the line y = 0. (a) Name three inputs that share the output 0. (b) Give one interval on which g is one-to-one.

Problem 16 Taxi Fare and Its Inverse ★★★

A taxi charges f(m) = 2.50 + 1.75m dollars for a ride of m miles. Write the inverse, say what it computes, and find the length of a ride that costs $23.50.

Problem 17 Lemonade Stand Revenue ★★★

A stand sells c(t) = 12t + 8 cups in t hours. Profit in dollars from c cups is r(c) = 1.75c − 20. Write r(c(t)), find the profit after 5 hours, and find the time needed to reach $162.

Problem 18 Spreading Ripple Area ★★★

A circular ripple has radius r(t) = 4 + 3t feet after t seconds, and area A(r) = πr². Write A(r(t)), find the area at t = 2, and find t when the area is 169π square feet.

Problem 19 Cabin Cost Split ★★★

A $480 cabin rental is split equally among n + 4 campers (n students plus 4 counselors), so each pays s(n) = 480n + 4 dollars. Find the inverse, describe what it gives, and find n when each camper pays $32.

Problem 20 Restrict, Invert and Evaluate ★★★

Let f(x) = 2x + 3 and g(x) = x² − 1. Find h = f ∘ g, restrict it to x ≥ 0, find h⁻¹, and evaluate h⁻¹(19).