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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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- Place Value in Decimal Number Systems
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- Order of Operations
- Some Useful Basic Numeracy
- Decimals
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- Exponents
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- Fractions
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- Reducing Radicals
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- Areas and Volumes of Figures
- Congruence and Similarity
- Functions
- Domain and Range of Functions
- Basics of Graphing
- Transformations
- Graphing Linear Functions
- Graphing Quadratic Functions
- Solving Systems of Linear Equations
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- Trigonometry
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- Exponential and Logarithmic Functions
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- Number Bases in Computer Arithmetic
- Linear Algebra
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- Basic Laws
- Place Value in Decimal Number Systems
- Decimals
- Significant Digits
- Prime Factorisation and Least Common Multiple
- Fractions
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- Ratios and Proportions
- Exponents
- Metric Conversions
- Reducing Radicals
- Factoring
- Solving Linear Equations
- Solving Quadratic Equations
- Polynomial Long Division
- Exponential and Logarithmic Functions
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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. If we draw a graph with the x-coordinate as the horizontal values, and y-coordinate as the vertical values. Then the **domain **is where the function lies horizontally and the **range** is where the function lies vertically.

In the graph above, notice 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 a [ bracket was used before -5 to state that it is greater than or equal to. A circle bracket, ( ), is used for open ends like infinity, or when it is strictly greater than.

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: Aug 19, 2022 3:18 PM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
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