Function Notation - Definition & Examples

(Video) Write Equations in Function Notation

by CK-12 Foundation

This video by CK-12 Foundation covers function notation and converting an equation into slope-intercept form.

$\large{\text{Summary}}$

When an equation is already in slope intercept form it is very simple to convert it into function notation. Suppose we have: $$y=2x+4$$ To convert it into function notation we just replace the $y$ with $f(x)$. $$f(x)=2x+4$$ This form easily informs us that whatever we input for $\color{#8925AE}{x}$ it will impact what $\color{#29A603}{f(x)}$ equals. $${\color{#29A603}{f(x)}}=2{\color{#8925AE}{x}}+4$$

Suppose we get an equation not in slope intercept form. $$2x+4y=18$$ To turn this into function notation, we need to first rewrite it by isolating $y$. $$\begin{align} 2x+4y&=18\\ 2x\,{\color{#8925AE}{-\,2x}}+4y&=18\,{\color{#8925AE}{-\,2x}}\\ 4y&=18-2x\\ \frac{4y}{\color{#8925AE}{4}}&=\frac{18-2x}{\color{#8925AE}{4}}\\ y&=\frac92-\frac12x \end{align}$$ Now we can convert it into function notation by replacing the $y$ with $f(x)$. $$f(x)=\frac92-\frac12x$$

Isolating Variables for Function Notation

Isolating variables is a very important skill in Algebra. This is because it is easier to read an interpret equations when the variable is isolated.

It's common in science to have a graph with a dependent and independent variable in a weird equation where you'll need to isolate the variable.

For example, look at the graph below.

This function is definied as $2y=x-6$. Let's isolate $y$ to get the slope intercept form.

$$\begin{align} 2y&=x-6\\ \frac{2y}{2} &= \frac{x}{2} - \frac62\\ y &= \frac12 - 3 \end{align}$$

Notice that this equation has variables on both sides. But, these are the same equation once we divide both sides by $2$.

However, we can read $y=\frac{1}{2}x-3$ and understand a few things very quickly. We know the slope is $\frac{1}{2}$ and the $y$-intercept is $-3$.

Since $y$ in this equation is isolated and written in terms of $x$, we can say that this equation is a function of $x$. We can replace $y$ with $f(x)$ to denote this.

$$y=\frac12 - 3\\ f(x)=\frac12 - 3$$

The slope and $y$-intercept are found more easily once the variable is isolated, or the function is in function notation. What if we don't have a given equation? We can use our given information to find linear equations.

If you have a set of ordered pairs, you can calculate the slope using points or using the graph. To find the function, you will need to plug in the values into one of the standard equations. The functions can be represented using slope intercept form, point slope form, or standard form.

Typically, you will find real numbers with these equations. You also may be asked to find the domain and range of a function. The domain of the function is all of its $x$ values, and the range of the function is all of its $y$ values. Using function notation also makes it easier to perform operations like adding and subtracting functions.

Be careful if you're asked to isolate the variable in a quadratic function! It gets really tricky there!

Function Notation

When solving functions, it is a lot easier to understand and to evaluate a function where there is only a variable on the left side. That variable is typically $y$, which we understand as a function of the other variable, typically $x$.

Therefore, we can write a function in function notation as $f(x)$.

So, if we want to say that $y$ is a function of $x$, we can rewrite that as $f(x)=...$

Function notation allows us to see an equation in terms of one variable. Let's look at the following equation:

$$3x+y=3$$

There's a lot going on in this equation, so it may seem initially hard to tackle. If we isolate the variable $y$, we can say that:

$$y=-3x+4$$

Since we understand that for every input of $x$, there will be an output of $y$, we know that $y$ is a function of $x$. Function notation allows us to say that:

$$f(x)=-3x+4$$

In more complex math, function notation is often used when writing piecewise definitions of functions. Furthermore, function notation is used when expressing and solving for an inverse function.

Let's do a practice problem!

$$3y=6x+9$$

Divide all terms by $3$. $$y=2x+3$$ Since we know that $y$ is a function of $x$, we can say that: $$f(x)=2x+3$$ Therefore, the second answer choice is correct.