Notes
What is a function?
- A function gives exactly one output for each input. Use the vertical line test on graphs and check for repeated inputs in ordered pairs.
- Function notation: $f(x)$ is the output for input $x$. Evaluate by substituting: $f(x) = 3x - 5 \Rightarrow f(4) = 7$.
Domain and range
- Domain = allowed inputs; range = outputs. Context can limit the domain (for example, whole numbers only).
Key features and rate of change
- Intercepts, increasing or decreasing intervals, and maximums or minimums describe a graph.
- Average rate of change from $a$ to $b$ is $\displaystyle \frac{f(b) - f(a)}{b - a}$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
A function needs exactly one output per input, and 2 maps to 4 and 5.
$-2(4) + 3(2) = -8 + 6 = -2$.
$f(-2) = f(2) = 4$, $f(-1) = f(1) = 1$ and $f(0) = 0$, so the range is $\{0, 1, 4\}$.
$f(0) = 9$. Setting $-3x + 9 = 0$ gives $x = 3$.
$f(4) = 32$ and $f(0) = 0$, so $\displaystyle \frac{32 - 0}{4 - 0} = 8$.
At $t = 0$, $A = 50$. The coefficient $-4$ is a decrease of 4 gallons per minute.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A taxi fare in dollars is $F(m) = 2.50 + 1.50m$, where $m$ is the number of miles traveled.
(a) Find $F(6)$ and explain what it means.
Show the answer and scoring guide
$F(6) = 2.50 + 1.50(6) = 11.50$. A 6-mile ride costs $11.50.
Check off the points you earned:
(b) Rides are at most 20 miles. State a reasonable domain and the range.
Show the answer and scoring guide
Domain: $0 \leq m \leq 20$. Range: $F(0) = 2.50$ and $F(20) = 2.50 + 30 = 32.50$, so $2.50 \leq F \leq 32.50$.
Check off the points you earned:
(c) Find the average rate of change of $F$ from 4 miles to 10 miles and interpret it.
Show the answer and scoring guide
$F(4) = 8.50$ and $F(10) = 17.50$. The rate is $\displaystyle \frac{17.50 - 8.50}{10 - 4} = \frac{9}{6} = 1.50$ dollars per mile.
Check off the points you earned: