Functions and Graphs
2.3 Domain and Range
Domain First
When a formula is given, the domain comes from values that keep the expression meaningful.
Definition: Domain and Range
The domain is the set of allowed input values.
The range is the set of resulting output values.
Fact: Common domain restrictions
- Denominators cannot be zero.
- Even roots require nonnegative radicands.
- Logarithm inputs must be positive.
Example: Domain of a rational function
Find the domain of
$$f(x)=\frac{3}{x-5}. $$
Show Solution
Denominator cannot be zero:
$$x-5\ne0\Rightarrow x\ne5.$$
Domain:
$$(-\infty,5)\cup(5,\infty).$$
Example: Domain of a radical function
Find the domain of
$$g(x)=\sqrt{2x+7}. $$
Show Solution
Require radicand to be nonnegative:
$$2x+7\ge0\Rightarrow x\ge-\frac72.$$
Domain:
$$\left[-\frac72,\infty\right).$$
MyOpenMath: Practice: domain restrictions (placeholder)
Range from Structure
Sometimes range is easiest from graph shape or transformed parent functions.
Example: Find range of a quadratic
Find the range of
$$h(x)=(x-1)^2-4.$$
Show Solution
This is an upward-opening parabola with vertex $(1,-4)$.
Minimum output is $-4$, so range is
$$[-4,\infty).$$