# Function Notation and Linear Functions

## Key Questions

• First, convert the linear equation to slope-intercept form. The slope-intercept form of a linear equation is: $y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and $\textcolor{b l u e}{b}$ is the y-intercept value.

Then switch out the $y$ variable for $f \left(x\right)$:

$f \left(x\right) = m x + b$

A function is a set of ordered pairs (points) formed from a defining equation, where, for each $x$-value there is only one $y$-value.

#### Explanation:

$x - - - - - - \to y$ represents a function

This means that you can choose an $x$-value and plug it into an equation, usually given as:

$y = \ldots . . \text{ " or" } f \left(x\right) = \ldots . .$

This will give you a $y$-value.

In a function there will be only ONE possible answer for $y$.

If you find you have a choice, then the equation does not represent a function.

The following are functions:

$y = - 3$

$y = 3 x - 5$

$y = 2 {x}^{2} - 3 x + 1$

$\left(1 , 2\right) , \left(2 , 2\right) , \left(3 , 2\right) , \left(4 , 2\right)$

The following are NOT functions:

$x = 3$

$y = \pm \sqrt{x + 20}$

• We can do more than giving an example of a linear equation: we can give the expression of every possible linear function.

A function is said to be linear if the dipendent and the indipendent variable grow with constant ratio. So, if you take two numbers ${x}_{1}$ and ${x}_{2}$, you have that the fraction $\frac{f \left({x}_{1}\right) - f \left({x}_{2}\right)}{{x}_{1} - {x}_{2}}$ is constant for every choice of ${x}_{1}$ and ${x}_{2}$. This means that the slope of the function is constant, and thus the graph is a line.

The equation of a line, in function notation, is given by $y = a x + b$, for some $a$ and $b \setminus \in \setminus m a t h \boldsymbol{R}$.