Functions
Functions (Extended)
- A function turns each input into one output: $f(x) = 3x - 5$, so $f(2) = 1$.
- the domain = allowed inputs; the range = possible outputs.
- The inverse $f^{-1}$ undoes $f$: write $y = f(x)$, swap, make $x$ the subject. For $f(x) = 3x - 5$, $f^{-1}(x) = \dfrac{x+5}{3}$.
Practice
For f(x) = 3x − 5, what is f(2)?
f(2) = 3(2) − 5 = 6 − 5 = 1.
Practice
The inverse of f(x) = 3x − 5 is f⁻¹(x) = (x + 5)/3. What is f⁻¹(1)?
f⁻¹(1) = (1 + 5)/3 = 6/3 = 2.
Composite functions
- $gf(x)$ means "do $f$ first, then $g$".
- If $f(x) = 2x$ and $g(x) = x + 3$: $gf(x) = 2x + 3$ and $fg(x) = 2(x+3) = 2x + 6$.
Practice
If f(x) = 2x and g(x) = x + 3, what is gf(x) at x = 5? (gf means do f first)
gf(5) = g(f(5)) = g(10) = 10 + 3 = 13.
You've got it
Key idea
- $f(x) = 3x - 5 \Rightarrow f(2) = 1$; domain = inputs, range = outputs
- inverse $f^{-1}$ undoes $f$ (swap and solve): $f^{-1}(x) = \dfrac{x+5}{3}$
- composite $gf(x)$ = do $f$ first, then $g$