Find the Range of \(f(x)=\dfrac{x+2}{|x+2|}\)

Solution

Given

\[ f(x)=\frac{x+2}{|x+2|} \]

We consider two cases depending on the sign of \(x+2\).

Case 1: \(x+2>0\)

That is,

\[ x>-2 \]

Then,

\[ |x+2|=x+2 \]

Therefore,

\[ f(x)=\frac{x+2}{x+2}=1 \]

Case 2: \(x+2<0\)

That is,

\[ x<-2 \]

Then,

\[ |x+2|=-(x+2) \]

Therefore,

\[ f(x)=\frac{x+2}{-(x+2)}=-1 \]

Restriction

At

\[ x=-2 \]

denominator becomes zero, so the function is not defined there.

Hence the function can take only two values:

\[ -1 \quad \text{and} \quad 1 \]