Piecewise-defined Function....1

[mathdad]

Can someone explain what exactly is a piecewise-defined function? How do we find the x and y intercepts of a piecewise-defined function?

y = x^3 if x is < or = 0...this is the upper piece

y = | x | if x > 0...this is the bottom piece

 

[MarkFL]

Can someone explain what exactly is a piecewise-defined function? How do we find the x and y intercepts of a piecewise-defined function?

y = x^3 if x is < or = 0...this is the upper piece

y = | x | if x > 0...this is the bottom piece

The given function is:

$$y(x)=\begin{cases}x^3, & x\le0 \\[3pt] |x|, & 0<x \\ \end{cases}$$

This tells us that on the interval $(-\infty,0]$, we have:

$$y(x)=x^3$$

And on the interval $(0,\infty)$, we have:

$$y(x)=|x|$$

Finding the intercepts is done the same way as for "ordinary" functions. To find the $x$-intercept(s), we set $y=0$ and solve for $x$, and to find the $y$-intercepts we take the point $(0,y(0))$. What are the intercepts for the given piecewise-defined function?

 

[mathdad]

y = {x}^{3}, when x = 0, y = 0.

0 = {x}^{3}, we take the cube root on both sides and get x = 0 and y = 0.

y = |0|, when x = 0, y = 0.

For 0 = |x|, do I square both sides?

 

[MarkFL]

y = {x}^{3}, when x = 0, y = 0.

0 = {x}^{3}, we take the cube root on both sides and get x = 0 and y = 0.

y = |0|, when x = 0, y = 0.

For 0 = |x|, do I square both sides?

We see that by the definition of the function, when $x=0$, then we have only $y(0)=0^3=0$. The absolute value piece is only defined for $0<x$. So, the $y$ intercept is at the origin, and this is the only place where $y=0$, and so the only intercept is at the origin. :D

 

[mathdad]

I am going to post a few more similar questions. I am not too clear in terms of y = |x|.