QuestionGraph $f(x)=\frac{|x+3|}{x+3}$ for $x$ in the range $[-4,4]$ .

Studdy Solution

STEP 1

Assumptions1. We are sketching the graph of the function $f(x)=\frac{|x+3|}{x+3}$ . . We are sketching the graph on the interval $[-4,4]$ .

STEP 2

The function $f(x)=\frac{|x+|}{x+}$ can be split into two cases, depending on whether $x+$ is positive or negative.1. When $x+ \geq0$ , $|x+| = x+$ , so $f(x)=1$ .
2. When $x+ <0$ , $|x+| = -(x+)$ , so $f(x)=-1$ .

STEP 3

The point where $x+3=0$ is $x=-3$ . This is the point where the function changes from $-1$ to $1$ .

STEP 4

Now, we can sketch the graph. For $x \in [-4,-3)$ , $f(x)=-1$ . For $x \in (-3,4]$ , $f(x)=1$ . At $x=-3$ , the function is not defined.

STEP 5

To make the sketch clearer, we can mark the points $(-4,-1)$ , $(-3,-1)$ , $(-3,1)$ , and $(4,1)$ on the graph. Draw a solid line from $(-4,-1)$ to $(-3,-1)$ , and another solid line from $(-3,1)$ to $(4,1)$ . Leave a hole at $(-3,-1)$ and $(-3,1)$ to indicate that the function is not defined at $x=-3$ .
The graph of $f(x)=\frac{|x+3|}{x+3}$ on the interval $[-4,4]$ has been sketched.

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