Functions and mappings

In a nutshell

A mapping shows how a set of inputs can be 'mapped' to a set of outputs. There are different types of mappings you need to know. For any function, the domain indicates what the inputs or $$x$$ values can take, while the range indicates what the outputs or $$y$$ values can take.

Functions and mappings

A function takes an input $$x$$​, transforms it by some mathematical operation before giving an output $$y$$. A mapping tells you whether one or multiple inputs give one or multiple outputs. If a mapping is one-to-one or many-to-one, then it is defined as a function.

	A one-to-one mapping, there is one value of B for each value of A. This is a function.
	A many-to-one mapping, more than one value of A can transform into the same value of B. This is a function.
	A many-to-many mapping, because one value of A can transform into multiple values of B or multiple values of A can transform into one value of B. This is not a function.

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Example 1

Describe the mappings for the following graphs. Which one are functions?

​$$y=2^x$$​​	​$$y= \tan (x)$$​​	​$$x^2+y^2=1$$​​

The first mapping is a one-on-one mapping because there is only one $$y$$ for each valour of $$x$$. This is a function.

The second mapping is a many-to-one mapping because different $$x$$ can lead to the same $$y$$. This is a function.

The last mapping is many-to-many and is not a function because there is more than one $$y$$ for the same $$x$$.

Domain and range

The domain shows the set of all inputs or $$x$$ values and the range shows the set of all outputs or $$y$$ values.

Example 2

Find the domain and the range of this function:

$$f(x)=x^2-3$$​

Any real number can be squared, so the domain of this function is $$x\in \R$$. However, the output of this function is more limited, because any real number raised to an even power is always positive, so the range is $$[-3,+\infty)$$ or $$y\ge -3$$.

Domain: $$\underline{x\in \R}$$, Range: $$\underline{y\ge -3}$$​

Cartesian plane

When a function is represented in the Cartesian plane, the domain is all the numbers that exist in the $$x$$-axis and the range is all the numbers that exist in the $$y$$-axis.