3.1: Function Definition

Definition: Function

A function is a rule that assigns to each element in the set of input values (the domain), one and only one element in the set of output values (the range).

Example 4.1.1

Determine if each of the following equations are functions:

1. $y = x^2 + 1$
2. $y^2 = x + 1$

Solution

1. To see the result of this equation, let x = 3.

$\begin{aligned} y &= x^2 + 1 \\ y &= 3^2 + 1 \\ y &= 9 + 1 \\ y &= 10\end{aligned}$

Any value entered for $x$ yields exactly one value for $y$.

There is only one solution for $y$, $y = 10$.

$y = x^2 + 1$ is a function!

2. To see the result of this equation, once again let $x = 3$.

$\begin{aligned} y^2 &= x + 1 \\ y^ 2 &= 3 + 1 = 4 \\ y &= \sqrt{4 } \\ y &= 2 \text{ or } y = −2\end{aligned}$

Any value entered for $x$ will not yield exactly one value for $y$. There are two solutions for $y$, $y = 2$ and $y = −2$.

$y^2 = x + 1$ is NOT a function!