Range Formula

The range of a function is the set of all actual output values that the function can produce for inputs in its domain.

The Formula

Range(f)={f(x)โˆฃxโˆˆDom(f)}\text{Range}(f) = \{f(x) \mid x \in \text{Dom}(f)\}

When to use: The range is the set of all possible "answers" the function can give โ€” some output values may be unreachable no matter what valid input you choose.

Quick Example

f(x)=x2f(x) = x^2 has range yโ‰ฅ0y \geq 0 (squares are never negative). f(x)=sinโก(x)f(x) = \sin(x) has range [โˆ’1,1][-1, 1].

Notation

Range(f)\text{Range}(f) or Im(f)\text{Im}(f) denotes the range (image). Written in set or interval notation: [0,โˆž)[0, \infty).

What This Formula Means

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 โ€” some output values may be unreachable no matter what valid input you choose.

Formal View

Im(f)={yโˆˆYโˆฃโˆƒโ€‰xโˆˆX:f(x)=y}\text{Im}(f) = \{y \in Y \mid \exists\, x \in X: f(x) = y\}

Worked Examples

Example 1

easy
Find the range of f(x)=x2+1f(x) = x^2 + 1.

Answer

[1,โˆž)[1, \infty)

First step

1
Since x2โ‰ฅ0x^2 \geq 0 for all real xx, the minimum value of x2x^2 is 00.

Full solution

  1. 2
    Therefore f(x)=x2+1โ‰ฅ0+1=1f(x) = x^2 + 1 \geq 0 + 1 = 1.
  2. 3
    As xโ†’ยฑโˆžx \to \pm\infty, f(x)โ†’โˆžf(x) \to \infty, so the range is [1,โˆž)[1, \infty).
For quadratics f(x)=ax2+bx+cf(x) = ax^2 + bx + c with a>0a > 0, the minimum value occurs at the vertex. Adding a constant shifts the range upward.

Example 2

medium
Find the range of g(x)=2x+1xโˆ’3g(x) = \frac{2x + 1}{x - 3} for xโ‰ 3x \neq 3.

Example 3

medium
Find the range of f(x)=9โˆ’x2f(x)=\sqrt{9-x^2}.

Common Mistakes

  • Copying the domain as the range - inputs and outputs are different sets; compute outputs separately.
  • Forgetting outputs a rule can never produce - check for squares, absolute values, and exponentials that block negatives.
  • Reporting one peak value instead of the full set - the range is an interval or set, not a single number.

Why This Formula Matters

Range tells you what answers a model can ever give: a profit function with range yโ‰ค100y\le 100 caps your best case. Confusing it with the domain swaps inputs for outputs and inverts the whole question. Recognizing it by "Which output values does the function actually reach as xx runs over its domain?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from domain and codomain and maximum value in a mixed problem set.

Frequently Asked Questions

What is the Range formula?

The range of a function is the set of all actual output values that the function can produce for inputs in its domain.

How do you use the Range formula?

The range is the set of all possible "answers" the function can give โ€” some output values may be unreachable no matter what valid input you choose.

What do the symbols mean in the Range formula?

Range(f)\text{Range}(f) or Im(f)\text{Im}(f) denotes the range (image). Written in set or interval notation: [0,โˆž)[0, \infty).

Why is the Range formula important in Math?

Range tells you what answers a model can ever give: a profit function with range yโ‰ค100y\le 100 caps your best case. Confusing it with the domain swaps inputs for outputs and inverts the whole question. Recognizing it by "Which output values does the function actually reach as xx runs over its domain?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from domain and codomain and maximum value in a mixed problem set.

What do students get wrong about Range?

The procedure for range is the easy part; the trap is copying the domain as the range. Asking "Which output values does the function actually reach as xx runs over its domain?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Range formula?

Before studying the Range formula, you should understand: function definition, domain.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Functions and Graphs: Complete Foundations for Algebra and Calculus โ†’
โ† Back to Range See All Examples Practice Problems

Related Concepts

FunctionDomain