Defining and Writing Functions

HOW TO

Given a function in equation form, write its algebraic formula.

1. Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable.

2. Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.

Example 9

Finding an Equation of a Function

Express the relationship $2 n+6 p=12$ as a function $p=f(n)$, if possible.

Solution

To express the relationship in this form, we need to be able to write the relationship where $p$ is a function of $n$, which means writing it as $p=[$ expression involving $n$].

$2 n+6 p=12$

$\begin{array}{ll}6 p=12-2 n & \text { Subtract } 2 n \text { from both sides. } \\p=\dfrac{12-2 n}{6} & \text { Divide both sides by } 6 \text { and simplify. } \\p=\dfrac{12}{6}-\dfrac{2 n}{6} & \\p=2-\dfrac{1}{3} n &\end{array}$

Therefore, $p$ as a function of $n$ is written as

$p=f(n)=2-\dfrac{1}{3} n$

Analysis

It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.

Example 10

Expressing the Equation of a Circle as a Function

Does the equation $x^{2}+y^{2}=1$ represent a function with $x$ as input and $y$ as output? If so, express the relationship as a function $y=f(x)$.

Solution

First we subtract $x^{2}$ from both sides.

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

We now try to solve for $y$ in this equation.

$\begin{aligned}y &=\pm \sqrt{1-x^{2}} \\&=+\sqrt{1-x^{2}} \quad \text { and }-\sqrt{1-x^{2}}\end{aligned}$

We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function $y=f(x)$.

Try It #6

If $x-8 y^{3}=0$, express $y$ as a function of $x$.

Q&A

Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?

Yes, this can happen. For example, given the equation $x=y+2^{y}$, if we want to express $y$ as a function of $x$, there is no simple algebraic formula involving only $x$ that equals $y$. However, each $x$ does determine a unique value for $y$, and there are mathematical procedures by which $y$ can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for $y$ as a function of $x$, even though the formula cannot be written explicitly.