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Functions and Readiness Checkpoint

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

This checkpoint reviews Algebra II readiness skills around function notation, domain and range, transformations, inverses, and composition.

What you'll learn

Why it matters: This checkpoint rehearses everything in the Functions unit: notation, domain, range, inverses, composition, and transformations. Every later Algebra II topic — quadratics, polynomials, exponentials, logs — treats these as the assumed vocabulary.

Worked example

Problem. If f(x) = 2x + 3 and g(x) = x - 5, find f(g(9)).

  1. Evaluate inside first: g(9) = 4.
  2. Now find f(4).
  3. 2(4) + 3 = 11.

Answer: 11

Practice problems

1. If f(x) = 4x - 1, find f(6).

Show solution
  1. Function Notation: First identify exactly what the question is asking: If f(x) = 4x - 1, find f(6).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. 4(6) - 1 = 23.
  4. Check the result by substituting or estimating: the response should match 23 and make sense in the original problem.

Answer: 23

2. For points (1, 4), (2, 6), and (3, 6), the range is...

Choices: {4, 6} · {1, 2, 3} · {1, 4} · {2, 3, 6}

Show solution
  1. Domain and Range: First identify exactly what the question is asking: For points (1, 4), (2, 6), and (3, 6), the range is...
  2. For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
  3. Range uses output values.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: {4, 6}

3. For f(x) = 1/(x - 8), what value is excluded?

Show solution
  1. Domain and Range: First identify exactly what the question is asking: For f(x) = 1/(x - 8), what value is excluded?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The denominator cannot be zero.
  4. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

4. y = f(x + 2) shifts the graph...

Choices: Left 2 · Right 2 · Up 2 · Down 2

Show solution
  1. Transformations: First identify exactly what the question is asking: y = f(x + 2) shifts the graph...
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Inside changes feel backward.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Left 2

5. If f(x) = x + 12, find f^-1(20).

Show solution
  1. Inverses: First identify exactly what the question is asking: If f(x) = x + 12, find f^-1(20).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Undo adding 12.
  4. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

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