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<0\\7 - x & \text{if }xgeq0end{cases}
choose the correct graph of f(x) below.
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.
discuss the behavior of f at its point(s) of discontinuity.

Explanation:

Step1: Analyze left - hand limit

For \(x < 0\), \(f(x)=-1 + x\). The left - hand limit as \(x\to0\) is \(\lim_{x\to0^{-}}f(x)=\lim_{x\to0^{-}}(-1 + x)=-1\).

Step2: Analyze right - hand limit

For \(x\geq0\), \(f(x)=7 - x\). The right - hand limit as \(x\to0\) is \(\lim_{x\to0^{+}}f(x)=\lim_{x\to0^{+}}(7 - x)=7\).

Step3: Analyze function value at \(x = 0\)

\(f(0)=7-0 = 7\). Since \(\lim_{x\to0^{-}}f(x)=-1\) and \(\lim_{x\to0^{+}}f(x)=7\), the function is discontinuous at \(x = 0\).

Step4: Graph the function

For \(y=-1 + x\) when \(x < 0\), the \(y\) - intercept is \(-1\) and the slope is \(1\). For \(y = 7 - x\) when \(x\geq0\), the \(y\) - intercept is \(7\) and the slope is \(- 1\). The correct graph will have a break at \(x = 0\).

Answer:

The correct graph is the one where the line \(y=-1 + x\) for \(x < 0\) and \(y = 7 - x\) for \(x\geq0\) has a break at \(x = 0\).
A. The point(s) of discontinuity is/are \(x = 0\).
At \(x = 0\), the left - hand limit \(\lim_{x\to0^{-}}f(x)=-1\), the right - hand limit \(\lim_{x\to0^{+}}f(x)=7\) and \(f(0)=7\). The function has a jump discontinuity at \(x = 0\) since the left - hand and right - hand limits exist but are not equal.