This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

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Different functions have different domains and ranges. Sometimes there are restrictions on domains and ranges.

The **domain** is the set of all input values. The **range** is the set of all output values.

When we draw graphs, we usually have the x-coordinates as the horizontal values, and the y-coordinates as the vertical values.

The **domain **is where the function lies horizontally and the **range** is where the function lies vertically.

In the graph above, notice how there are no values in the domain less than -5 on the left side of the graph. On the right side, the graph keeps going.

If we call this graph \(f(x)\), the **domain **is stated \(D: \{x\in \mathbb{R}|x \geq -5\}\). This means all x values (input values) can be any Real Number greater than or equal to -5. You can also state the domain as the following \(D:[-5, \infty) \). Notice that a [ bracket (square bracket) was used before -5 to state that it is greater than or equal to. Open/rounded brackets, ( ), are used for open ends like infinity, or when a group of values does not include an endpoint. For example, stating that \( x > 10 \) means x is any value bigger than 10, but not 10 itself.

The **range **of the graph \(f(x)\) above, is stated as \(R: \{f(x)\in \mathbb{R}|f(x) \leq 5\}\). You can use \(y\) instead of \(f(x)\). This means that all \(y\) or \(f(x)\) values are less than or equal to 5. In the other notation, it is stated \(R:(-\infty, 5]\)

Knowing the properties and graphs of functions helps you find the domain and range.

For example, we know that \(f(x) = x^2\) is a parabola opening up with vertex at \((0,0)\).

The **domain** of \(f(x) \) is all the numbers, so \(D:\{x\in\mathbb{R}\}\).

The **range** is all the numbers greater than or equal to 0, so \(R:\{f(x)\in\mathbb{R}|f(x) \geq 0\}\).

Other quadratic functions will have similar domains and ranges.

For example, if \(f(x) = 3x^2-10\) is the same graph that has been stretched vertically by a factor of 3 and moved down 10 units.

The domain is still \(D:\{x\in\mathbb{R}\}\).

The range is now all the numbers greater than or equal to -10, since the vertex has moved down 10 units, so \(R:\{f(x)\in\mathbb{R}|f(x) \geq -10\}\).

So knowing the base function properties helps you determine domains and ranges after the graph has been transformed (dilated, moved. reflected).

Here are other common functions:

The **root **function, \(f(x) = \sqrt{x}\): \(D:\{x\in\mathbb{R}|x \geq 0\}\) and \(R:\{f(x)\in\mathbb{R}|f(x) \geq 0\}\)

The **cubic **function, \(f(x) = x^3\): \(D:\{x\in\mathbb{R}\}\) and \(R:\{f(x)\in\mathbb{R}\}\)

The **reciprocal **function \(f(x) = \frac{1}{x}\): \(D:\{x\in\mathbb{R}|x \neq 0\}\) and \(R:\{f(x)\in\mathbb{R}|x \neq 0\}\)

Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.

- Last Updated: Mar 4, 2024 10:18 AM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
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