Algebra 2 Composition and Inverse Functions Test with Point Values (PDF)

Algebra 2 math test: Composition and Inverse Functions – Maths PDF

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This composition inverses test for Algebra 2 covers composition and inverse functions (Common Core F-BF.A.1c, F-BF.B.4a), with a free PDF to print.

This 45-minute test on composition and inverse functions has 9 questions worth 20 points in total. The questions move from direct skills to multi-step problems. Show your work, because partial credit is given for a correct setup and sound reasoning.

Time and points

About this test

Grade: Algebra 2
Chapter: Composition and Inverse Functions
Standards: F-BF.A.1c, F-BF.B.4a
Time: 45 minutes
Total: 20 points

Composition inverses test: questions

Question 1 – Compose from tables [2 pts]

Values of f and g.

Use the table to find \(f(g(1))\) and \(g(f(2))\).

Question 2 – Compose expressions [3 pts]

Let \(f(x) = x^{2} – 1\) and \(g(x) = 2x + 3\). Find \(f(g(x))\) and \(g(f(x))\) in simplified form, and decide whether they are equal.

Question 3 – Domain [2 pts]

Let \(f(x) = \sqrt{x}\) and \(g(x) = x – 5\). Find \(f(g(x))\) and its domain.

Question 4 – Linear inverse [2 pts]

Find the inverse of \(f(x) = \frac{x – 4}{5}\).

Question 5 – Rational inverse [3 pts]

Find the inverse of \(f(x) = \frac{2x + 1}{x – 3}\).

Question 6 – Verify [2 pts]

Verify that \(f(x) = \sqrt[3]{x + 7}\) and \(g(x) = x^{3} – 7\) are inverses.

Question 7 – Graph of an inverse [2 pts]

Graph of f and a second line.

The graph of \(f(x) = \frac{x}{2} + 1\) is shown with a second line. The point \((4, 3)\) is on \(f\). Give the corresponding point on \(f^{-1}\) and find \(f^{-1}(x)\).

Question 8 – Restricted domain [2 pts]

f(x) = x² – 1 for x ≥ 0.

Find the inverse of \(f(x) = x^{2} – 1\) for \(x \ge 0\) and its domain.

Question 9 – Find the error [2 pts]

Mei says \(f(g(x)) = g(f(x))\) for every pair of functions. Use \(f(x) = x + 1\) and \(g(x) = 2x\) to show that she is wrong.

Key facts behind this composition inverses test

Key facts

f(g(x)): apply g first, then f.
Composition order matters.
Inverse: swap x and y, then solve for y.
f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.
Graphs of f and f^(-1) are mirror images over y = x.

How to use this composition inverses test

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

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