Unsure what to do with a particular written form of a function f(x)

[Ajroberts]

It says that f(x) equals both tan pi x divided by 4, and 1, I dont understand what that means.

 

[Dr.Peterson]

This is called "piecewise definition"; you should look up that phrase in the index of your textbook.

Here is an explanation of it: https://www.mathsisfun.com/sets/functions-piecewise.html

Here's another: https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGTypePiecewise.html

You'll find the continuity aspect near the bottom of this page: https://www.mathsisfun.com/calculus/continuity.html

 

[Steven G]

View attachment 13984
It says that f(x) equals both tan pi x divided by 4, and 1, I dont understand what that means.

Please look at the link which Professor Peterson supplied.
What you say is not exactly true. f(x) equals tan pi x divided by 4 ONLY if |x|<1 and f(x) = 1 only if |x|> 1

What exactly does |x|<1 mean? And what does |x|> 1 mean?

 

[pka]

View attachment 13984
It says that f(x) equals both tan pi x divided by 4, and 1, I dont understand what that means.

\(\displaystyle f(x)=\begin{cases}\tan\left(\dfrac {x\pi}{4}\right) &\text{if } |x|<1 \\1 &\text{if } |x|\ge 1\end{cases}\)
Have a look at this plot.
From the plot we see \(\displaystyle \mathop {\lim }\limits_{x \to - {1^ + }} f(x) = - 1\)
BUT \(\displaystyle \bf{\mathop {\lim }\limits_{x \to - {1^ - }}} f(x) = 1\) from the function definition (if \(\displaystyle x<-1\) then \(\displaystyle |x|>1\)).