Question

graph f, locate all points of discontinuity, and discuss the behavior of f at these points.
f(x)=(\begin{cases}2 + x & \text{if }xleq1\\4 - x & \text{if }x>1end{cases}
graph f. choose the correct graph 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.

Explanation:

Step1: Analyze left - hand limit

For \(x\leq1\), \(f(x)=2 + x\). The left - hand limit as \(x\to1\) is \(\lim_{x\to1^{-}}f(x)=\lim_{x\to1^{-}}(2 + x)=2 + 1=3\).

Step2: Analyze right - hand limit

For \(x>1\), \(f(x)=4 - x\). The right - hand limit as \(x\to1\) is \(\lim_{x\to1^{+}}f(x)=\lim_{x\to1^{+}}(4 - x)=4-1 = 3\).

Step3: Analyze function value at \(x = 1\)

When \(x = 1\), \(f(1)=2+1 = 3\). Since \(\lim_{x\to1^{-}}f(x)=\lim_{x\to1^{+}}f(x)=f(1)=3\), the function is continuous everywhere.

Step4: Graph the function

For \(y = 2 + x\) when \(x\leq1\), the \(y\) - intercept is 2 and the slope is 1. For \(y = 4 - x\) when \(x>1\), the \(y\) - intercept is 4 and the slope is - 1. The correct graph has a line \(y = 2 + x\) for \(x\leq1\) and \(y = 4 - x\) for \(x>1\) meeting at the point \((1,3)\). The correct graph is B.

Answer:

Graph: B
Discontinuity: B. There are no points of discontinuity.