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Functions, composite and inverse
A function is a rule that turns an input into an output. Functions can be combined and reversed.
Function notation
f(x) = 3x + 1 means: multiply by 3, then add 1.
f(4) = 3 × 4 + 1 = 13
f(4) = 3 × 4 + 1 = 13
Composite functions
fg(x) means do g first, then f.
f(x) = 3x + 1 and g(x) = x2: fg(2) = f(4) = 13. gf(2) = g(7) = 49.
f(x) = 3x + 1 and g(x) = x2: fg(2) = f(4) = 13. gf(2) = g(7) = 49.
Inverse functions
f-1(x) undoes f. Write y = f(x), swap x and y, and rearrange.
f(x) = 3x + 1 → x = 3y + 1 → y = (x − 1)3
f-1(x) = (x − 1)3
f(x) = 3x + 1 → x = 3y + 1 → y = (x − 1)3
f-1(x) = (x − 1)3
f(x) = 2x − 5. Find f-1(9).
- f-1(x) = (x + 5)2
- f-1(9) = 142 = 7
Answer: 7
f(a) means put a into the rule. fg(x) means g first, then f. For the inverse, swap x and y and rearrange.
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