Function Notation Review
A free Algebra II lesson from the “Functions and Notation” unit, with a worked example and practice problems including step-by-step solutions.
Function notation names a rule. In f(x), x is the input and f(x) is the output. To evaluate a function, substitute the input everywhere the variable appears.
What you'll learn
- Evaluate functions
- Interpret function notation
- Use inputs and outputs in context
Why it matters: Tax brackets, shipping rate tables, and fuel-economy charts use function notation so that the rule and the value are unambiguous. Reading f(7) as 'the output when the input is 7' keeps a calculation honest no matter how complex the rule is.
Worked example
Problem. If f(x) = 3x - 4, find f(7).
- Substitute 7 for x.
- f(7) = 3(7) - 4.
- 21 - 4 = 17.
Answer: 17
Practice problems
1. If f(x) = 2x + 5, find f(3).
Show solution
- Warm-up: First identify exactly what the question is asking: If f(x) = 2x + 5, find f(3).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Substitute 3.
- 2(3) + 5 = 11.
- Check the result by substituting or estimating: the response should match 11 and make sense in the original problem.
Answer: 11
2. If g(x) = x^2 - 4, find g(6).
Show solution
- Warm-up: First identify exactly what the question is asking: If g(x) = x^2 - 4, find g(6).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- 6^2 - 4 = 32.
- Check the result by substituting or estimating: the response should match 32 and make sense in the original problem.
Answer: 32
3. If h(t) = 3t - 1, find h(-2).
Show solution
- Core Practice: First identify exactly what the question is asking: If h(t) = 3t - 1, find h(-2).
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- 3(-2) - 1 = -7.
- Check the result by substituting or estimating: the response should match -7 and make sense in the original problem.
Answer: -7
4. In f(4) = 9, the output is...
Choices: 9 · 4 · f · 13
Show solution
- Core Practice: First identify exactly what the question is asking: In f(4) = 9, the output is...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The function value is the output.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 9
5. If p(x) = x^2 + 2x and p(a) = 15 when a = 3, what is p(3)?
Show solution
- Challenge: First identify exactly what the question is asking: If p(x) = x^2 + 2x and p(a) = 15 when a = 3, what is p(3)?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- The notation p(a) means evaluate the rule at a.
- Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.
Answer: 15
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