Explanation of function $|y| = x$

Question

First time posting here.

Studying this book and this statement came across.

In the equation $|y| = x$, $y$ is not a function of $x$ because every nonnegative $x$-value has two $y$-values. For example, if $x = 3$, $|y| = 3$ has the solutions $y = 3$ and $y = -3$.

Huettenmueller, Rhonda. College algebra demystified. New York: McGraw-Hill Professional, 2014.

So far, I never saw operations on $y$ side of the equation. How can I solve those ?

Thanks in advance

Answer 1

This is the inverse of $y = |x|$ so you can visualize it by reflecting it over the line $y=x$. If you reason through it, you can observe that $|y| = x$ basically means that $x$ is always positive since the absolute value function will always give you a positive value and that $|y|$ can be any real number. This leads to this plot:

Observe that it is the inverse of $y = |x|$ and that $y$ has a range of $(-\infty, \infty)$ while the $x$ is always positive. This is not a function since it does not pass the vertical line test.

Answer 2

Let observe that by definition $|y|=x \implies x\ge 0$ and

• $y\ge 0 \implies |y|=x \iff y=x$

• $y< 0 \implies |y|=x \iff -y=x \iff y=-x$

then we have two different function both defined for not negative $x$ values.

Plot of |y|=x

Answer 3

The graph of $x=|y|$ doesn't pass the vertical line test.

Answer 4

First solve the equation for y:- $$|y| = x$$ So $$y=\pm x$$ Now let y = f(x) Therefore, $$f(x)=\pm x$$ We know by definition of function that every element of its domain should have only one image but here we can see that $f(x)$ has two values for all $x > 0$, one positive and one negative, so it is not a function.