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.
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 \(ln(x)\).
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.
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.
A function is a type of relation.
A 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.
If you must simplify, then (2) would be your 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(x) &= \sqrt{2(9x+3)-1} \\ h(x) &= \sqrt{18x+6-1} \\ h(x) &= \sqrt{18x+5} \end{align}
Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.