Question

Please explain how this function behaves when it approaches the given x values!

Answer 1

Answer and Step-by-step explanation:

This is a piecewise function because it is defined by more than two functions. Basically, we want to take the limit here. Recall that if a function $f(x)$  approaches some value $L$  as $x$  approaches $a$  from both the right and the left, then the limit of $f(x)$ exists and equals  $L$. Here we won't calculate the limit, but apply some concepts of it. So:

a. $as \ x \rightarrow +\infty, \ k(x) \rightarrow +\infty$

Move on the x-axis from the left to the right and you realize that as x increases y also increases without bound.

b. $as \ x \rightarrow -\infty, \ k(x) \rightarrow 0$

Move on the x-axis from the right to the left and you realize that as x decreases to negative values y approaches zero.

c. $as \ x \rightarrow 2, \ k(x) \rightarrow 0$

Since the function is continuous here, we can say that $k(2)=0$

d. $as \ x \rightarrow -2, \ k(x) \rightarrow 0$

The function is discontinuous here, but $k(-2)$ exists and equals 0 as the black hole indicates at $x=-2$.

e. $as \ x \rightarrow -4, \ k(x) \rightarrow 2$

The function is also discontinuous here, but the black hole indicates that this exists at $x=-4$, so $k(-4)=2$

f. $as \ x \rightarrow 0, \ k(x) \rightarrow 4$

Since the function is continuous here, we can say that $k(0)=4$