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Functions

A List of Lessons in the Functions Curriculum

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What Is a Function?

Functions express a relationship between an input and an output. If you put a number in to a function, another number will come out.

Dictionary Definition

The Merriam-Webster dictionary defines a function as "a mathematical correspondence that assigns exactly one element of one set to each element of the same or another set."
A function can be shown as a mapping diagram, where each number in a set of inputs is mapped to a number in a set of outputs. The mapping diagram shown below represents a function. Notice that there is a predictable relationship between the input and the output. In the example, 1 is added to each input:

functions mapping diagram

Functions As Equations

Functions are usually expressed as an equation, where the relationship between the input and output is shown. The equation below shows a function, which will be called f.

functions function notation

The equation shows that the output of the function, which is denoted f(x) (said as "f of x"), is related to the input of the function, which is denoted x. The function takes the input x and adds 1 to it.

The Curriculum

The lessons are grouped into mini-curriculum to help you organise your learning. A brief description is given for each mini-curriculum. Click the MORE button to learn more.

Functions

functions about functions

Functions relate inputs to outputs. An equation is usually used to describe this relationship. Functions can be thought of as a mapping. An input is mapped to an output. If you put a number in, the function relates it to another number. In this mini-curriculum, you will learn about functions.

Function
A function is a relation between a set of inputs and a set of outputs, such that each input is related to exactly one output.
Here is a function that takes an input and adds 1 to it. It is shown as a mapping diagram (left) and as an equation (right):
functions function

Evaluate a Function
To evaluate a function, put an input into a function and see what output it relates to.
Here is an example of evaluating a function. The function f(x) = x + 1 is evaluated at x = 2:
functions evaluate function



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functions
how to evaluate a function

Plotting Functions

functions plot

Functions can be plotted on a graph. The graph shows the relationship between the input (along the horizontal x-axis) and the output (up the vertical y-axis). In this mini-curriculum, you will learn about plotting functions.

Find Y-Intercept
To find the y-intercept of a funciton, find where the graph of the function crosses the y-axis.
This is done by evaluating the function at x = 0.
functions plot y-intercept

Find X-Intercepts
To find the x-intercepts of a funciton, find where the graph of the function crosses the x-axis.
This is done by finding the values of x which make f(x) = 0. The method will be different for different functions.
functions plot x-intercepts

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how to find the y-intercept of a function
how to find the x-intercepts of a function

Composite Functions

functions about composite functions

Composite functions combine functions, so that the output of one function becomes the input of another. In this mini-curriculum, you will learn about composite functions.

Composite Function
A composite function is a function of a function. It combines two or more functions so that the output of one function becomes the input of another.
Here is a composite function. The first function takes an input and adds 1 to it. This is passed as an input into a second function, which doubles it. This is shown as a mapping diagram (left) and as an equation (right):
functions composite functions

Find a Composite Function
To find a composite function, pass one function into another.
functions find composite function

Evaluate a Composite Function
To evaluate a composite function, find the composite function and evaluate it.
functions evaluate composite function

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composite functions
how to find a composite function
how to evaluate a composite function

Inverse Functions

functions about inverse functions

The inverse of a function reverses that function. If a function relates an input to an output, the inverse function relates the output back to the input. In this mini-curriculum, you will learn about inverse functions.

Inverse Function
An inverse function is a function that reverses another function.
Here is an example of an inverse function. It relates the outputs of a function back to the inputs
functions inverse function

Find an Inverse Function
To find the inverse of a function, rearrange a function to find what the input is in terms of the output.
functions find inverse function

Find an Inverse Function Using a Graph
To find the inverse of a function using a graph, reflect the graph of the function in the line y = x (a diagonal line).
functions find inverse function using graph

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inverse functions
how to find an inverse function
how to find an inverse function using a graph

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This page was written by Stephen Clarke.

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