Question

graph f, locate all points of discontinuity, and discuss the behavior of f at these points.
f(x)=\begin{cases}-1 + x&\text{if }x < - 1\\5 - x&\text{if }xgeq - 1end{cases}
choose the correct graph of f(x) below.
a.
b.
c.
d.
locate all points of discontinuity. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the point(s) of discontinuity is/are x =
(use a comma to separate answers as needed.)
b. there are no points of discontinuity.

Explanation:

Step1: Check left - hand limit at $x = - 1$

For $x\lt - 1$, $f(x)=-1 + x$. Then $\lim_{x
ightarrow - 1^{-}}f(x)=\lim_{x
ightarrow - 1^{-}}(-1 + x)=-1+( - 1)=-2$.

Step2: Check right - hand limit at $x = - 1$

For $x\geq - 1$, $f(x)=5 - x$. Then $\lim_{x
ightarrow - 1^{+}}f(x)=\lim_{x
ightarrow - 1^{+}}(5 - x)=5-( - 1)=6$.

Step3: Evaluate function at $x=-1$

$f(-1)=5-( - 1)=6$.

Step4: Determine continuity

Since $\lim_{x
ightarrow - 1^{-}}f(x)=-2$, $\lim_{x
ightarrow - 1^{+}}f(x)=6$ and $f(-1) = 6$, the left - hand limit is not equal to the right - hand limit at $x=-1$. So the function is discontinuous at $x = - 1$.

Step5: Analyze the graph

For $y=-1 + x$ when $x\lt - 1$, it is a line with slope $1$ and $y$-intercept $-1$. For $y = 5 - x$ when $x\geq - 1$, it is a line with slope $-1$ and $y$-intercept $5$. The correct graph will have a break at $x=-1$.

Answer:

Locate all points of discontinuity: A. The point(s) of discontinuity is/are $x=-1$.
(For the graph part, without seeing the actual graph details in full - resolution, we can't choose from A, B, C, D precisely, but the above analysis helps in identifying the correct graph which should have a break at $x = - 1$)