Functions

Definitions

A function is like a machine that follows a specific rule. For every number you put in, you get exactly one number out.
Let's imagine a machine whose rule is "multiply by 2".

If we put in a 3, we get out a 6. If we put in a 5, we get out a 10. A table of values helps us organize these pairs:

Input	3	5	8	10
Output	6	10	16	20

To work with these rules more efficiently, mathematicians developed a special notation.
To represent this machine, we write $\textcolor{olive}{f}(\textcolor{colordef}{\text{input}}) = \textcolor{colorprop}{\text{output}}$. The parentheses $($ $)$ indicate the action of the function $\textcolor{olive}{f}$ on the input.
We use function notation to name functions and their variables, replacing "$\textcolor{colordef}{\text{input}}$" by "$\textcolor{colordef}{x}$" and "$\textcolor{colorprop}{\text{output}}$" by "$\textcolor{colorprop}{f(x)}$".
We can represent a function in the following way:

For example, if the rule is "twice the input":

we have $\textcolor{olive}{f}(\textcolor{colordef}{x}) = \textcolor{colorprop}{2x}$.

When the input is $\textcolor{colordef}{x} = \textcolor{colordef}{1}$, we get:$$\begin{aligned}\textcolor{olive}{f}(\textcolor{colordef}{1}) &= 2 \times \textcolor{colordef}{(1)}\\ &= \textcolor{colorprop}{2}\end{aligned}$$

Definition Function

A function, $f$, is a rule that assigns to each input value from a set called the domain, exactly one output value in a set called the range.

$f$ is the name of the function (the rule).
$x$ is the input variable.
$f(x)$ is the output value when the input is $x$. It is read as "$f$ of $x$".
$f(x)$ is called the image of $x$ under $f$.

Example

The function $f$ is defined by the rule $f(x)=2x-1$. Find the value of $f(5)$.

Answer

To find $f(5)$, we substitute the input value $x=5$ into the function's rule:$$\begin{aligned}[t]f(x) &= 2x - 1 \\ f(5) &= 2(5) - 1 \\ &= 10 - 1 \\ &= \boldsymbol{9}\end{aligned}$$

Tables of Values

Definition Table of Values

A table of values is a table that organizes the relationship between the input values ($x$) and their corresponding output values ($f(x)$) for a function.

Example

Complete the table of values for the function $f(x)=x^2$.

$x$	$-2$	$-1$	$0$	$1$	$2$
$f(x)$

Answer

We substitute each value of $x$ into the function $f(x)=x^2$:

$\begin{aligned}[t] f(-2) &= (-2)^2 \\ &= 4 \end{aligned}$
$\begin{aligned}[t] f(-1) &= (-1)^2\\ &= 1 \end{aligned}$
$\begin{aligned}[t] f(0) &= (0)^2 \\ &= 0 \end{aligned}$
$\begin{aligned}[t] f(1) &= (1)^2 \\ &= 1 \end{aligned}$
$\begin{aligned}[t] f(2) &= (2)^2 \\ &= 4 \end{aligned}$

The completed table is:

$x$	$-2$	$-1$	$0$	$1$	$2$
$f(x)$	$4$	$1$	$0$	$1$	$4$

Graphs of Functions

While a table of values is useful for listing some input–output pairs of a function, a graph is a powerful tool for visualizing how the output changes when the input changes. A graph gives us a picture of the function's behavior.

Definition Graph of a Function

The graph of a function $f$ is the set of all points with coordinates $(\textcolor{colordef}{x}, \textcolor{colorprop}{f(x)})$ in a coordinate plane. The input value, $x$, is plotted on the horizontal axis (the x-axis), and the output value, $f(x)$, is plotted on the vertical axis (the y-axis). When the function is defined for all $x$ in an interval, we can connect these points to form the curve of the function.

Method Plotting a Graph from a Table

To plot the graph of a function from its table of values:

Draw a coordinate plane with a suitable scale on each axis and label the axes.
For each pair $(x, f(x))$ in the table, plot the corresponding point on the coordinate plane.
If the function is defined for all $x$ in the interval shown, connect the points with a straight line or a smooth curve.

Example

Plot the graph of the function $f(x) = x - 1$ using its table of values.

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$-3$	$-2$	$-1$	$0$	$1$	$2$

Answer

We plot the points $(-2, -3)$, $(-1, -2)$, $(0, -1)$, $(1, 0)$, $(2, 1)$, and $(3, 2)$ from the table. These points lie on the same straight line, so we connect them to draw the graph of $f(x)=x-1$.

Finding Outputs from Inputs on a Graph

Method Finding the Value of $f(x)$ from a Graph

To find the output $f(x)$ for a given input $x$ using a graph:

Locate the input value on the horizontal x-axis.
Move vertically from that point until you reach the curve of the function.
Move horizontally from the intersection point to the vertical y-axis and read the corresponding value. This y-value is the output $f(x)$.

Example

Using the graph of the function $f$ below, find the value of $f(2)$.

Answer

We follow the graphical method:

Start at $x=2$ on the horizontal axis.
Move up to meet the curve.
Move horizontally to the vertical axis and read the value, which is $3$.

Therefore, $\boldsymbol{f(2) = 3}$.

Finding Inputs from Outputs

We have learned how to take an input ($x$) and find its output ($f(x)$). Now, we will learn how to work backwards: if we know the output, can we find the input(s) that produced it? This process is called finding the preimage(s) (or input(s)) of a given value.

Method Finding Inputs from Outputs on a Graph

To find the preimage(s) of a value $y$ (i.e., find all $x$ such that $f(x)=y$):

Locate the output value $y$ on the vertical y-axis.
Draw a horizontal line from this value across the graph.
Find the intersection point(s) where the horizontal line crosses the function's curve.
Move vertically to the x-axis from each intersection point to read the corresponding input value(s). These are the preimages.

Example

Using the graph of the function $f$ below, find $x$ such that $f(x)= 3$.

Answer

We apply the graphical method:

We locate $y=3$ on the vertical axis.
We draw a horizontal line at $y=3$.
The line intersects the curve at two points.
We move vertically from these points to the x-axis to read the values, which are $-2$ and $2$.

The preimages of $3$ are $\boldsymbol{-2}$ and $\boldsymbol{2}$.

Finding a preimage graphically is useful for visualization, but for an exact answer, we can use algebra.

Method Finding Inputs from Outputs Algebraically

To find the preimage(s) of a value $y$ for a function $f(x)$:

Set the function's formula equal to the output value: $\boldsymbol{f(x) = y}$.
Solve the resulting equation for $x$.

Example

Let $f(x) = 3x + 12$. Find $x$ such that $f(x)=0$.

Answer

We need to find the value of $x$ such that $f(x)=0$. We set up the equation and solve:$$\begin{aligned} f(x) &= 0 \\ 3x + 12 &= 0 \\ 3x &= -12 && \color{gray}\text{(subtract 12 from both sides)} \\ x &= \frac{-12}{3} &&\color{gray}\text{(divide both sides by 3)} \\ x &= -4\end{aligned}$$The preimage of $0$ is $\boldsymbol{x = -4}$.
Check: $f(-4) = 3(-4) + 12 = -12 + 12 = 0$. The answer is correct.

\(x\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(f(x)\)

\(x\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(f(x)\)	\(4\)	\(1\)	\(0\)	\(1\)	\(4\)

\(x\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(f(x)\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)