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.

Explanation:

Step1: Analyze left - hand limit

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

Step2: Analyze right - hand limit

For \(x\geq - 1\), \(f(x)=5 - x\). The right - hand limit as \(x\to - 1\) is \(\lim_{x\to - 1^{+}}(5 - x)=5-( - 1)=6\).

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

\(f(-1)=5-(-1) = 6\) since \(x=-1\) satisfies \(x\geq - 1\).

Step4: Check continuity

Since \(\lim_{x\to - 1^{-}}f(x)=-2
eq\lim_{x\to - 1^{+}}f(x) = 6\), the function is discontinuous at \(x=-1\).

Step5: Analyze graph behavior

For \(x < - 1\), the graph of \(y=-1 + x\) is a line with slope \(m = 1\) and \(y\) - intercept \(-1\). For \(x\geq - 1\), the graph of \(y = 5 - x\) is a line with slope \(m=-1\) and \(y\) - intercept \(5\). The graph has a break at \(x=-1\).

The correct graph is the one where the line \(y=-1 + x\) approaches \(y=-2\) as \(x\) approaches \(-1\) from the left and the line \(y = 5 - x\) starts at the point \((-1,6)\) and has a negative slope for \(x\geq - 1\).

Answer:

Without seeing the actual graphs, we can't pick a letter - option. But the graph should have a line \(y=-1 + x\) for \(x < - 1\) and a line \(y = 5 - x\) starting at the point \((-1,6)\) for \(x\geq - 1\) with a discontinuity at \(x=-1\).