Example 10 from Algebra and Trigonometry, 3.1 Functions and Function Notation

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 $y = f (x) .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

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

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

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

$\begin{array}{l} y = \pm \sqrt{1 - x^{2}} \\ = + \sqrt{1 - x^{2}} and - \sqrt{1 - x^{2}} \end{array}$
We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function $y = f (x) .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

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

How did it go?

Solution

Your turn

Solution

Your turn