Range Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Range.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
The range of a function is the set of all actual output values that the function can produce for inputs in its domain.
The range is the set of all possible "answers" the function can give — some output values may be unreachable no matter what valid input you choose.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Range is what the function actually produces, not what's theoretically possible.
Common stuck point: Range is often harder to find than domain—may need graphing.
Sense of Study hint: Sketch the graph or make a table of outputs for several inputs, then look for the lowest and highest y-values the function actually reaches.
Worked Examples
Example 1
easySolution
- 1 Since x^2 \geq 0 for all real x, the minimum value of x^2 is 0.
- 2 Therefore f(x) = x^2 + 1 \geq 0 + 1 = 1.
- 3 As x \to \pm\infty, f(x) \to \infty, so the range is [1, \infty).
Answer
Example 2
mediumPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyExample 2
hardBackground Knowledge
These ideas may be useful before you work through the harder examples.