# Math help from the Learning Centre

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.

## What is a Function?

You can think of a function like a machine. You input something into the machine and the function will output something.

For example, the function may be a machine that triples every number. Another example may be the function $$x^2 +1$$. The $$x$$ represents whatever you input into the function-- or in other words, whatever number you give to the machine. This function takes every input and squares it, then adds 1 to it.

## Function Notation

A function can be named anything, the most common function names are $$f$$ and $$g$$. There is an input inside the function that can also be denoted by any letter, but most commonly it is denoted by $$x$$.

Thus, $f(x)$

means that the name of the function is $$f$$ and the input is $$x$$. The output is what it is equal to.

$f(x)=x^2$

is read "f of x equals x squared" where $$x^2$$ is the output.

Another example $g(A)=A+2$

is a function that is named $$g$$, with input $$A$$, and the output adds all inputs by 2 to get $$A+2$$.

There are also special functions with specific names. For example: $$sin(x)$$ and $$ln(x)$$.

## Domain and Range

All possible values (or elements) that can be inputted into a function belong to what we call the domain. Whereas all possible elements that can be outputted belong to what we call the range or codomain.

• In other words, the domain of a function is the set of all possible inputs.
• The range/codomain of a function is the set of all possible outputs.

Example

If you think of a Vending Machine as a function: the domain would be money while the range is all the available snacks/drinks inside.

## Functions vs. Relations

A function is a type of relation.

relation between two sets is officially described as a collection of ordered pairs containing one element from each group. Think of a relation as a way of pairing values together in a specific way.

Example

$$y = 2x+ 1$$ is a relation.

If $$x = 1$$, then $$y = 2(1) + 1 = 3$$, meaning that the ordered pair $$(2,3)$$ is a member of this relation.

Notice how the ordered pair is expressed as: $$(input , output )$$.

Every function is a relation, but NOT every relation is a function.
A relation is a function if every input has only one possible output. The output does not have to be unique.

Example

Let's say that this diagram represents a relation with domain (set of inputs) X and range (set of outputs) Y. This is a function because every element in the domain X has one unique output.

Example

Let's say that this diagram represents a relation with domain (set of inputs) X and range (set of outputs) Y. This is not a function because the element 2 in the domain has more than one output.

Example 1

If $$f(x) = 2x^2 -1$$, find $$f(3)$$.

Solution

$$f(3)$$ means that for the function named $$f$$, which represents $$2x^2 -1$$, the input is 3 or $$x=3$$. In other words, substitute $$x=3$$ into $$2x^2 -1$$ and find the output.

\begin{align} f(x) &= 2x^2-1 \\ \\ f(3) &= 2(3)^2 - 1 \\ &= 2(9) - 1 \\ &= 18 - 1 \\ f(3) &= 17\end{align}

Example 2

If $$g(y) = 9y +3$$, find $$g(-3)$$.

Solution

$$g(-3)$$ means to substitute $$y=-3$$ into $$g(y) = 9y +3$$.

\begin{align} g(y) &= 9y+3 \\ \\ g(-3) &= 9(-3)+3 \\ &= -27 + 3 \\ g(-3) &= -24\end{align}

Example 3

If $$h(x) = \sqrt{2x-1}$$, find $$h(A)$$.

Solution

$$h(A)$$ means to substitute $$x=A$$ into $$h(x) = \sqrt{2x-1}$$.

\begin{align} h(x) &= \sqrt{2x-1} \\ \\ h(A) &= \sqrt{2A-1}\end{align}

Example 4

If $$f(x) = 2x^2 -1$$, find $$f(x+1)$$.

Solution

$$f(x+1)$$ means to substitute $$x+1$$ into $$f(x) = 2x^2-1$$.

\begin{align} f(x) &= 2x^2-1 \\ \\ f(x+1) &= 2(x+1)^2 - 1 \qquad \qquad &\text{(1)}\\ &= 2(x^2+2x+1) - 1 \\ &= 2x^2+4x+2-1 \\ f(x+1)&= 2x^2+4x+1 &\text{(2)} \end{align}

If you are not required to simplify, then (1) is fine as a final answer.

Example 5

If $$g(y) = 9y +3$$, find $$g(y^2)$$.

Solution

$$g(y^2)$$ means to substitute $$y^2$$ into $$g(y) = 9y +3$$.

\begin{align} g(y) &= 9y+3 \\ \\ g(y^2) &= 9(y^2)+3 \\ g(y^2) &= 9y^2+3\end{align}

Example 6

If $$h(x) = \sqrt{2x-1}$$, find $$h(9x+3)$$.

Solution

$$h(9x+3)$$ means to substitute $$9x+3$$ into $$h(x) = \sqrt{2x-1}$$.

\begin{align} h(x) &= \sqrt{2x-1} \\ \\ h(9x+3) &= \sqrt{(9x+3)} \\ h(9x+3) &= \sqrt{9x+3} \end{align}