Precalculus / Unit 1: Functions and Their Graphs
1.6 Piecewise Functions and Rate of Change
Notes
A piecewise function uses different rules on different parts of the domain.
- Pick the rule whose condition the input satisfies.
- The average rate of change of $f$ from $x = a$ to $x = b$ is $\displaystyle \frac{f(b) - f(a)}{b - a}$, the slope of the secant line.
- Example: $f(x) = x^2 - x$ from $x = 1$ to $x = 4$: $\displaystyle \frac{12 - 0}{3} = 4$.
Practice
Let $f(x) = x + 1$ for $x < 2$ and $f(x) = x^2 - 3$ for $x \ge 2$. What is $f(2) + f(0)$?
$f(2) = 4 - 3 = 1$ and $f(0) = 0 + 1 = 1$, so the sum is 2.
What is the average rate of change of $f(x) = x^2 - x$ from $x = 1$ to $x = 4$?
$f(4) = 12$ and $f(1) = 0$, so $\displaystyle \frac{12 - 0}{4 - 1} = 4$.
What is the average rate of change of $f(x) = 3x^2$ from $x = 0$ to $x = 2$?
$\displaystyle \frac{f(2) - f(0)}{2} = \frac{12 - 0}{2} = 6$.
Ready for more? Take the Unit 1 test.