General functions
Function operations and composition worksheets
Add, subtract, multiply, divide and compose polynomial functions. Choose numbers, variables or mixed inputs, with ten questions and a separate worked answer key.
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Inverse functions worksheets
Identify inverse pairs or find inverse functions in twelve original questions. Practice linear functions and powers with explicit domains, exact radicals and worked answers.
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Use the printed domain. For an inverse pair, check both compositions. To find an inverse, swap x and y, solve for y and swap the original domain and range.
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Undo the rule. Keep the domain.
Choose twelve inverse-pair questions, twelve find-the-inverse questions, or a mixture with six of each. Practice fractional and integer linear rules, shifted powers and scaled powers.
Every even-power function has a printed domain restriction. It selects the left or right branch and determines the sign of the inverse root. The answer key states the inverse domain and range, with algebraic steps for finding an inverse and exact counterexamples for incorrect pairs.
True inverse pairs undo each other in both orders on the stated domains. Testing a few numbers can help detect a mistake, but the algebra establishes the identity for every allowed input.
Learning objective
What this activity builds
Reverse a function rule and use its domain and range to decide whether the inverse is a function.
Worked example
Find the inverse of f(x) = 3(x − 2)² for x >= 2.
x = 3(y − 2)², so y = 2 + √(x/3). The inverse has domain x >= 0 and range y >= 2.
The restriction selects the right half of the parabola, so y − 2 is nonnegative after swapping the variables. On the left half, x <= 2, the inverse would instead be 2 − √(x/3).
Teaching notes
Make the practice count
- Start with twelve find-the-inverse questions before mixing in identification.
- Read f⁻¹ as inverse function, not the reciprocal 1/f.
- Keep fractions and radicals exact rather than rounding.
- Use a real cube root for negative inputs.
- An even power needs a one-to-one domain restriction.
- For a false inverse pair, one valid counterexample is enough.
- For a true inverse pair, numerical checks alone do not prove an identity on an entire domain.
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