# 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.

## Functions

### 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 $$x^2 +1$$. This function takes every input and squares it, then adds it by 1.

### A Common 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 special functions with a specific names for example $$sin(x)$$ and $$ln(x)$$.

### Function Rules

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.

Not every equation can be a function, a function must have the following:

• every element in the domain has a unique output in the range (called the one-to-one property)
• no input is left out (called the onto property)

In other words, one element cannot lead to two outputs in the range. This is a function because every element in the domain X has one unique output, and every element in domain X is used. No input is left out. This is not a function because the element 2 in the domain has more than one output and not every element in the domain is used (e.g., 3 and 4 have no outputs).

## Evaluating Functions

For $$f(x) = 2x^2 -1$$, $$g(y) = 9y +3$$, $$h(x) = \sqrt{2x-1}$$, find

1. $$f(3)$$
2. $$g(-3)$$
3. $$h(A)$$
4. $$f(x+1)$$
5. $$g(y^2)$$
6. $$h(9x+3)$$

$$f(3)$$ means that for the function named $$f$$, which is 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.

$$f(3) = 2(\textbf{3})^2 -1 = 2(\textbf{9}) - 1 = 17$$

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

$$g(-3) = 9(\textbf{-3}) +3 = 30$$

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

$$h(A) = \sqrt{2A-1}$$

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

$$f(x+1) =2(\textbf{x+1})^2-1$$

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

$$g(y^2) = 9(\textbf{y^2}) +3$$

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

$$h(A) = \sqrt{2(\textbf{9x+3})-1} = \sqrt{\textbf{18x+6}-1} = \sqrt{18x+5}$$