Question:
The range of the function $f(x)=\frac{x+2}{|x+2|}, x \neq-2$ is
(a) {−1, 1}
(b) {−1, 0, 1}
(c) {1}
(d) (0, ∞)
Solution:
(a) {−1, 1}
$f(x)=\frac{x+2}{|x+2|}, x \neq-2$
Let $y=\frac{x+2}{|x+2|}$
For $|x+2|>0$,
or $x>-2$,
$y=\frac{x+2}{x+2}=1$
For $|x+2|<0$,
or $x<-2$
$y=\frac{x+2}{-(x+2)}=-1$
Thus, $y=\{-1,1\}$
or range $f(x)=\{-1,1\}$.
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