Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
These problems cover operations on functions, composition, inverse functions, and the reflection of a graph across the line y = x. They follow Common Core standards F-BF.A.1b, F-BF.A.1c, and F-BF.B.4a.
Early problems build combined functions and evaluate compositions; later ones invert linear, root, rational, and restricted functions and use inverses in real contexts such as temperature, discounts, and taxi fares.
Standards: F-BF.A.1b, F-BF.A.1c, F-BF.B.4a · Level: Grade 11 Algebra 2
Problem 1 Adding, subtracting, and multiplying functions ★★★
For h(x) = 5x + 2 and k(x) = x2 − 6, find (h + k)(x), (h − k)(x), and (h · k)(x).
Problem 2 Evaluating combined functions ★★★
Using the same h and k, find (h + k)(3) and (h · k)(−2).
Problem 3 Composition at a number ★★★
Let h(x) = 4x − 7 and k(x) = x + 6. Find h(k(5)) and k(h(5)).
Problem 4 Composition as an expression ★★★
For h(x) = x2 + 3 and k(x) = x − 5, find h(k(x)) and k(h(x)).
Problem 5 A function machine ★★★
An input is processed by k(x) = 3x and then by h(x) = x − 4. The figure shows the machine for the input 5. Find the output for the input 8.
Problem 6 A linear inverse ★★★
Determine the inverse of h(x) = 4x + 9.
Problem 7 An inverse with a fraction ★★★
Determine the inverse of h(x) = 5x − 24.
Problem 8 Verifying inverses ★★★
Show by composition that h(x) = 6x + 1 and k(x) = x − 16 are inverses.
Problem 9 Reflecting a graph ★★★
The graph of h(x) = 3x − 2 and the line y = x are shown. Reflect the points (−1, −5), (0, −2), (1, 1), and (2, 4) of h across y = x, then write the equation of h−1.
Problem 10 A cube root inverse ★★★
Determine the inverse of h(x) = x3 + 6 and evaluate h−1(14).
Problem 11 A restricted domain ★★★
The function h(x) = x2 − 7 is restricted to x ≥ 0. Find h−1, its domain, and verify h(h−1(9)) = 9.
Problem 12 Inverse of a square root function ★★★
Determine the inverse of h(x) = √(x + 3), and state the domain of h−1.
Problem 13 Domain of a composition ★★★
For h(x) = √x and k(x) = x + 8, write h(k(x)) and k(h(x)), and give the domain of each.
Problem 14 Inverse of a table ★★★
A function h is given by the points (1, 6), (2, 9), and (3, 12). Write the points of h−1, find h−1(12), and use the figure to explain the relationship.
Problem 15 A rational inverse ★★★
Determine the inverse of h(x) = 5x + 3x − 2 and check it by computing h(h−1(x)).
Problem 16 Converting temperatures ★★★
A food scientist converts Celsius to Fahrenheit with F(c) = 95c + 32. Find the inverse function that converts Fahrenheit to Celsius, use it for 98.6°F, and verify by composition.
Problem 17 Order of discounts ★★★
A shopper has a $10-off coupon, C(p) = p − 10, and a 20%−off sale, D(p) = 0.8p. For a $60 jacket, find D(C(60)) and C(D(60)), and decide which order saves more money.
Problem 18 Taxi fare and miles ★★★
A taxi fare is C(m) = 2.5m + 3.5 dollars for m miles. Find the inverse function that gives miles from the fare, and use it to find the distance for a $21 fare. The figure shows the fare graph.
Problem 19 Making a function invertible ★★★
The function h(x) = x2 − 6x + 5 fails the horizontal line test. Restrict the domain to x ≥ 3, find h−1, and state its domain and range.
Problem 20 When do two compositions agree? ★★★
Let h(x) = 4x − 1 and k(x) = x2 + 3. There are exactly 2 values of x for which h(k(x)) = k(h(x)); find both.
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