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Functions Checkpoint

A free College Algebra lesson from the “Functions” unit, with a worked example and practice problems including step-by-step solutions.

This checkpoint samples function notation, domain and range restrictions, composition, inverse functions, and transformation readiness.

What you'll learn

Why it matters: This checkpoint connects the language of functions to valid inputs, outputs, composition, and inverse reasoning. A parent or student can read it as evidence that function notation is becoming a usable tool, not just a symbol pattern.

Worked example

Problem. If f(x) = x + 2 and g(x) = 4x, find f(g(3)).

  1. g(3) = 12.
  2. f(12) = 14.
  3. So f(g(3)) = 14.

Answer: 14

Practice problems

1. If f(x) = 2x - 5, find f(9).

Show solution
  1. Function Notation: First identify exactly what the question is asking: If f(x) = 2x - 5, find f(9).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. 18 - 5 = 13.
  4. Check the result by substituting or estimating: the response should match 13 and make sense in the original problem.

Answer: 13

2. For h(x) = 1/(x - 10), what value is excluded?

Show solution
  1. Domain: First identify exactly what the question is asking: For h(x) = 1/(x - 10), what value is excluded?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. x - 10 cannot be 0.
  4. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

3. If f(x) = x^2 and g(x) = x + 1, find f(g(4)).

Show solution
  1. Composition: First identify exactly what the question is asking: If f(x) = x^2 and g(x) = x + 1, find f(g(4)).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. g(4) = 5, then f(5) = 25.
  4. Check the result by substituting or estimating: the response should match 25 and make sense in the original problem.

Answer: 25

4. The inverse of f(x) = x + 6 is...

Choices: f^-1(x) = x - 6 · f^-1(x) = x + 6 · f^-1(x) = 6x · f^-1(x) = x/6

Show solution
  1. Inverses: First identify exactly what the question is asking: The inverse of f(x) = x + 6 is...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Undo adding 6.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: f^-1(x) = x - 6

5. For h(x) = sqrt(2x - 6), what is the smallest allowed x-value?

Show solution
  1. Domain: First identify exactly what the question is asking: For h(x) = sqrt(2x - 6), what is the smallest allowed x-value?
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. 2x - 6 must be at least 0.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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