Precalculus / Unit 1: Functions and Their Graphs
1.4 Combining and Composing Functions
Notes
You can add, subtract, multiply and divide functions. Composition feeds one function into another: $(f \circ g)(x) = f(g(x))$.
- Work from the inside out: compute $g(x)$ first.
- Order matters: usually $f \circ g \ne g \circ f$.
- For $\displaystyle \frac{f}{g}$, exclude any $x$ where $g(x) = 0$.
- Example: $f(x) = x + 2$, $g(x) = x^2$. Then $(f \circ g)(3) = f(9) = 11$ but $(g \circ f)(3) = g(5) = 25$.
Practice
If $f(x) = 2x - 1$ and $g(x) = x^2 + 3$, what is $(f \circ g)(x)$?
$f(g(x)) = 2(x^2 + 3) - 1 = 2x^2 + 5$.
If $f(x) = x + 2$ and $g(x) = x^2$, what is $(g \circ f)(3)$?
$f(3) = 5$, then $g(5) = 25$.
If $\displaystyle f(x) = \sqrt{x}$ and $g(x) = x - 4$, what is the domain of $(f \circ g)(x)$?
$\displaystyle (f \circ g)(x) = \sqrt{x - 4}$, which needs $x - 4 \ge 0$.
Ready for more? Take the Unit 1 test.