Precalculus / Unit 1: Functions and Their Graphs
1.1 Functions, Domain and Range
Notes
A function assigns each input exactly one output. The domain is the set of allowed inputs and the range is the set of outputs.
- Vertical line test: a graph is a function if no vertical line hits it twice.
- Domain restrictions: no division by zero, and no even root (like a square root) of a negative number.
- Example: $\displaystyle f(x) = \frac{\sqrt{x - 1}}{x - 4}$ needs $x \ge 1$ and $x \ne 4$, so the domain is $[1, 4) \cup (4, \infty)$.
- Evaluate by substituting: if $f(x) = x^2 - 3x$, then $f(-2) = 4 + 6 = 10$.
Practice
What is the domain of $\displaystyle f(x) = \frac{\sqrt{x - 3}}{x - 5}$?
The square root needs $x - 3 \ge 0$, and the denominator cannot be zero, so $x \ne 5$.
If $f(x) = 2x^2 - 3x + 1$, what is $f(-2)$?
$2(4) - 3(-2) + 1 = 8 + 6 + 1 = 15$.
What is the range of $g(x) = x^2 + 4$?
$x^2 \ge 0$, so $x^2 + 4 \ge 4$, and the value 4 is reached at $x = 0$.
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