The Life Starter
Sign up
Grade 12 · unit review

Review: Precalculus: Composition and Inverse Functions

Function operations must respect input order and domain. A composition feeds one output into another function, while an inverse reverses a one-to-one relationship and exchanges the original domain and range.

Download reviews and answer keys

Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Let f(x)=x² and g(x)=2x+1. Find (f∘g)(7).
Need a hint?
  • In f(g(x)), g acts on the supplied input before f acts on its result.
  • Read the innermost operation first.
2. Let f(u)=√u and g(x)=x−7. What is the real domain of f(g(x))?
Need a hint?
  • A composed function exists only when its inner function is defined and that output belongs to the outer function’s domain.
  • Checking the inner function’s domain alone is insufficient.
3. Find the inverse of f(x)=3x+7.
Need a hint?
  • A nonconstant linear function is one-to-one.
  • An inverse-function symbol is not a reciprocal exponent.
4. The function f(x)=x²+7 has domain x≥0. Find its inverse and domain.
Need a hint?
  • Squaring on all real inputs loses sign information and is not one-to-one.
  • Do not return both square-root branches as a function output.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Let f(x)=x² and g(x)=2x+1. Find (f∘g)(9). Candidate answers: (A) 360 (B) 361 (C) 362 (D) 722. 2. Let f(u)=√u and g(x)=x−9. What is the real domain of f(g(x))? Candidate answers: (A) Every real x (B) x≠9 (C) x≤9 (D) x≥9. 3. Find the inverse of f(x)=3x+9. Candidate answers: (A) f⁻¹(x)=(x+9)/3 (B) f⁻¹(x)=(x−9)/3 (C) f⁻¹(x)=1/(3x+9) (D) f⁻¹(x)=3x−9. 4. The function f(x)=x²+9 has domain x≥0. Find its inverse and domain. Candidate answers: (A) f⁻¹(x)=x²−9, x≥0 (B) f⁻¹(x)=±√(x−9), all real x (C) f⁻¹(x)=√(x+9), x≥0 (D) f⁻¹(x)=√(x−9), x≥9.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 361. Composition applies g first: g(9)=2(9)+1=19. Then apply f: (19)²=361. Reversing the order would give g(f(9)) instead. 2. x≥9. The output of g must be in the square-root domain, so x−9≥0. Thus x≥9, including the endpoint where the root is zero. 3. f⁻¹(x)=(x−9)/3. Set y=3x+9, then solve x=(y−9)/3. Swap the input/output labels. An inverse undoes a function; it is not generally its reciprocal. 4. f⁻¹(x)=√(x−9), x≥9. The nonnegative domain makes f one-to-one. Solve y=x²+9 with x≥0: x=√(y−9). The range of f, [9,∞), becomes the inverse domain.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.