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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.

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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.

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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.

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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 \(h(x) = \sqrt{2x-1}\)

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

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