Algebra 2
Functions and function notation
A function assigns exactly one output to each input. f(x) is read 'f of x' and names the output for input x — it is not multiplication. Domain is the set of allowed inputs; range is the set of resulting outputs.
Why it matters
Function notation is the language of precalculus and calculus. Students who misread f(x + 2) as f(x) + 2 lose marks repeatedly across transformations, composition and derivatives.
Worked example
If f(x) = x² − 3x, find f(−2) and f(a + 1)
- 1Substitute −2 everywhere x appears. (−2)² − 3(−2) = 4 + 6
- 2State the first value. f(−2) = 10
- 3Substitute a + 1 as a whole. (a + 1)² − 3(a + 1)
- 4Expand and simplify. a² + 2a + 1 − 3a − 3 = a² − a − 2
Answer: f(−2) = 10 and f(a + 1) = a² − a − 2
Common mistakes
- Treating f(x) as f times x.
- Expanding (a + 1)² as a² + 1.
- Confusing f(x) + 2 (vertical shift) with f(x + 2) (horizontal shift).
How to check your answer
- Use the vertical line test on a graph: two outputs for one input means it is not a function.
- Check the domain for values that would divide by zero or take the root of a negative.
Related topics
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