Functions and Relations
Learn easily with Video Lessons and Interactive Practice Problems
Contents
 Introduction
 What is a Function?
 Graphing Functions
 Special Functions
 Inverse Functions
 Function Operations
Introduction
An ordered pair is a set of inputs and outputs and represents a relationship between the two values. A relation is a set of inputs and outputs, and a function is a relation with one output for each input.
What is a Function?
Some relationships make sense and others don’t. Functions are relationships that make sense. All functions are relations, but not all relations are functions.
A function is a relation that for each input, there is only one output.
Here are mappings of functions. The domain is the input or the xvalue, and the range is the output, or the yvalue.
Each xvalue is related to only one yvalue.
Athough the inputs equal to 1 and 1 have the same output, this relation is still a function because each input has just one output.
This mapping is not a function. The input for 2 has more than one output.
Graphing Functions
Using inputs and outputs listed in tables, maps, and lists, makes it is easy to plot points on a coordinate grid. Using a graph of the data points, you can determine if a relation is a function by using the vertical line test. If you can draw a vertical line through a graph and touch only one point, the relation is a function.
Take a look at the graph of this relation map. If you were to draw a vertical line through each of the points on the graph, each line would touch at only one point, so this relation is a function.
Special Functions
Special functions and their equations have recognizable characteristics.
Constant Function
$f(x) = c$
The cvalue can be any number, so the graph of a constant function is a horizontal line. Here is the graph of $f(x) = 4$
Identity Function
$f(x) = x$
For the identity function, the xvalue is the same as the yvalue. The graph is a diagonal line going through the origin.
Linear Function
$f(x) = mx + b$
An equation written in the slopeintercept form is the equation of a linear function, and the graph of the function is a straight line.
Here is the graph of $f(x)= 3x +4$
Absolute Value Function
$f(x) = x$
The absolute value function is easy to recognize with its Vshaped graph. The graph is in two pieces and is one of the piecewise functions.
This is just a sample of the most common special functions.
Inverse Functions
An inverse function reverses the inputs with its outputs.
$f(x) = 3x  4$
Change the inputs with the outputs to create the inverse of this function.
$\begin{align} f(x) &= 3x 4\\ y &= 3x 4\\ x &= 3y 4\\ x +4 &= 3y 4 + 4\\ x+ 4&= 3y\\ \frac{x + 4}{3}&= \frac{3}{3}y\\ f^{1}(x)&=\frac{x + 4}{3} \end{align}$
The inverse of $f(x) = 3x  4$ is $f^{1}(x) =\frac{x + 4}{3}$.
Not every inverse of a function is a function, so use the vertical line test to check.
Function Operations
You can add, subtract, mutiply, and divide functions.
 $f(x) + g(x) = (f + g)(x)$
 $f(x)  g(x) = (f  g)(x)$
 $f(x) \times g(x) = (f \times g)(x)$
 $\frac{f(x)}{g(x)}= \frac{f}{g}(x)$
Look at two examples of function operations:
What is the sum of these two functions? Simply add the expressions.
$\begin{align} f(x) &= 2x + 3\\ g(x) &= 3x + 5\\ (f + g) (x) &= 2x + 3 + 3x + 5 = 5x + 8 \end{align}$
What is the product of these two functions? Simply multiply the expressions.
$\begin{align} f(x) &= x + 4\\ g(x) &= x + 7\\ (f\times g)(x) &= (x + 4) \times (x +7) = x^{2} + 11x + 28 \end{align}$
All Videos in this Topic
Videos in this Topic
Functions and Relations (7 Videos)
All Worksheets in this Topic
Worksheets in this Topic
Functions and Relations (7 Worksheets)

What is a Function? The Difference between Functions and Relations
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Function Operations
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Inverse Functions
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Graphing Functions
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Convert between Tables, Graphs, Mappings, and Lists of Points
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Linear and Nonlinear Functions
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Special Functions
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