What a function does
A function is a rule that turns each input value into exactly one output. We usually write it as f(x), read "f of x". To evaluate a function you substitute the given number in place of x and simplify.
Key idea
If f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11. Whatever replaces x on the left must replace x everywhere on the right.
Inverse and composite functions
The inverse f-1(x) reverses the effect of f. To find it, write y = f(x), swap the roles of x and y, then make y the subject. A composite function such as fg(x) means "do g first, then f": fg(x) = f(g(x)).
Worked example
Let f(x) = 2x + 1. Find f-1(x).
Write y = 2x + 1, so 2x = y - 1 and x = (y - 1)/2. Swapping symbols gives f-1(x) = (x - 1)/2.
Check: f(3) = 7 and f-1(7) = (7 - 1)/2 = 3. The inverse returns the original input.
Domain and range
The domain is the set of allowed inputs; the range is the set of possible outputs. A value that makes a denominator zero must be excluded from the domain. For f(x) = x2, every output is zero or positive, so the range is f(x) ≥ 0.
Remember
- fg(x) = f(g(x)): innermost function acts first.
- Composition is usually not commutative: fg(x) ≠ gf(x) in general.
- Applying f then f-1 returns the starting value.