Functions: Range

Understand what the range of a function is, learn about special ranges (like square root, quadratic, and periodic functions), and practice reading ranges from graphs.

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Learning Guide: The Coordinate System

Learning Guide: Range of a Function

The range of a function is the set of all possible output values (usually yy-values) obtained by evaluating the function over its entire domain.

Range of Square Root Functions

For the principal square root function f(x)=xf(x) = \sqrt{x}, the output is defined as the non-negative square root. Since we cannot obtain a negative number from a real square root, the range is restricted to non-negative numbers: y≥0y \ge 0 (or [0,∞)[0, \infty) in interval notation).
xyRange: y ≥ 0f(x) = √x
Figure 3a: The square root function f(x)=xf(x) = \sqrt{x} yields non-negative outputs (y≥0y \ge 0).

Range of the Function f(x)=x2f(x) = x^2

A quadratic function like f(x)=x2f(x) = x^2 takes any input and multiplies it by itself. Because the product of any real number with itself is never negative (e.g., (−3)2=9(-3)^2 = 9 and 02=00^2 = 0), the domain is all real numbers (R\mathbb{R}), but the outputs are always non-negative. Therefore, its range is y≥0y \ge 0 (or [0,∞)[0, \infty)).
xyRange: y ≥ 0Domain: ℝf(x) = x²
Figure 3b: The quadratic function f(x)=x2f(x) = x^2 yields non-negative outputs (y≥0y \ge 0).

Range of Periodic Functions

Periodic functions repeat their values in cycles. For these functions, the output values cycle between a specific minimum and maximum value, meaning their range is bounded and forms a closed interval: [ymin,ymax][y_{\text{min}}, y_{\text{max}}] (or ymin≤y≤ymaxy_{\text{min}} \le y \le y_{\text{max}}). For example, the sine function, f(x)=sin⁡(x)f(x) = \sin(x), cycles continuously, restricting its range to [−1,1][-1, 1] (don't worry if you aren't familiar with sin⁡(x)\sin(x) yet—it is only used here as an example).
xyMax: y = 1.5Min: y = -1.5Periodic waveRange: [-1.5, 1.5]
Figure 3c: The periodic wave oscillates between a minimum and maximum boundary.

Mastering SealMath

When answering range questions, you can enter inequalities directly using keyboard shortcuts. The editor automatically converts them to math symbols:
To WriteType on KeyboardMath Display
Greater than or equal to>=≥\ge
Less than or equal to<=≤\le
Greater than>>>
Less than<<<
Learning Topics

Frequently Asked Questions

Why does the x-coordinate always come first in an ordered pair?
By mathematical convention, coordinates are always written in alphabetical order as (x,y)(x, y). This standardized order ensures that anyone around the world can communicate and locate points on a coordinate plane consistently without ambiguity.
What makes a relation a function?
A relation is a function if and only if each input value is associated with exactly one output value. If a single input has multiple different outputs, it is not a function.
How do you find where a function crosses the y-axis?
To find where a function crosses the yy-axis, calculate f(0)f(0) by replacing xx with 00 in the function formula. The resulting point on the graph will be (0,f(0))(0, f(0)).
What happens if we input a value outside the domain?
If you input a value outside the domain, the function is undefined for that value. For example, in f(x)=1xf(x) = \frac{1}{x}, inputting x=0x = 0 results in division by zero, which has no defined mathematical value.
How can you identify the domain of a function from its graph?
To find the domain from a graph, look at the graph's horizontal extent along the xx-axis. Find the leftmost and rightmost points of the graph, taking note of whether the endpoints are solid (included) or open circles (excluded).
What is the difference between domain and range?
The domain is the set of all valid input values (usually xx) that you can feed into a function, while the range is the set of all output values (usually yy) that the function produces as a result.
How can you identify the range of a function from its graph?
To find the range from a graph, look at the graph's vertical extent along the yy-axis. Find the lowest and highest points of the graph, taking note of whether these endpoints are included (solid) or excluded (open circles).
What does it mean for a function to be continuous?
Intuitively, it means you can draw the function's graph without lifting your pencil. Formally, a function must be defined at the point, and the graph must connect smoothly without any gaps, jumps, or holes.
How do you find points where a function is not continuous?
Look for inputs that make the function undefined (like division by zero). For example, f(x)=x2−16x−4f(x) = \frac{x^2 - 16}{x - 4} is discontinuous at x=4x = 4 because you cannot divide by zero, creating a hole in the graph.
How do I determine whether a function is increasing or decreasing from its formula?
For linear functions f(x)=mx+bf(x) = mx + b, the function increases if the slope m>0m > 0 and decreases if m<0m < 0. For quadratic functions f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k, check the sign of aa: if a>0a > 0, the parabola opens up, so it decreases for x<hx < h and increases for x>hx > h.
Why can't I use y-values to define intervals of increase or decrease?
Although we look at the vertical rise or fall (y-values) to determine *what* the graph does, the intervals must specify *where* it happens horizontally. By convention, intervals of increase/decrease partition the domain of the function, which is represented by the xx-axis.