Key Idea: A composite function feeds one function into another. An inverse function undoes a function. You find both, state their domains, and explain what they mean in context.
Composite functions
- (f∘g)(x) = f(g(x)): do the inner g first, then f.
- Order matters: f∘g ≠ g∘f in general.
- Domain: x must be legal for g, and g(x) must be legal for f. Watch √ and division.
- Example: f(x) = √x and g(x) = x − 4 give √(x − 4), with x ≥ 4.
Inverse functions
- Find it — y = f(x) → swap x and y → solve for y.
- Graph it — Reflect f in the line y = x. The point (a, b) becomes (b, a).
- Check it exists — f must be one-to-one. If not, restrict the domain first, e.g. x ≥ 0 for x².
- Swap domain and range — The range of f becomes the domain of the inverse.
Composite: work from the inside out. Inverse: undo the function by reversing its input and output.
f(x) = 2x + 3. Find f⁻¹(x). (x − 3)/2.
Why must f(x) = x² be restricted before you invert it? Two inputs give the same output, so it fails the horizontal line test.
f(x) = x² + 1 for x ≥ 0. Where is f⁻¹ defined? For x ≥ 1, since f only outputs values of 1 or more.
What does f⁻¹(b) ask you to do? Solve f(x) = b.