Question

Number of distinct elements in the range of the function $f(x) = \dfrac{{x + 2}}{{|x + 2|}}$ is

Answer 1

Hint: To solve this question we will use the property of modulus function. First we will find from where to where the function is positive and where it is negative. Then will use that information to find the range of the given function.

Formula Used: $f(x) = |x|$
Then $f(x) = x\,\,\,\,if\,x \geqslant 0$
And $f(x) = - x\,\,\,\,\,if\,x \leqslant 0$

Complete step by step solution: Given, function is $f(x) = \dfrac{{x + 2}}{{|x + 2|}}$
Let $f(x) = |x|$
Then $f(x) = x\,\,\,\,if\,x \geqslant 0$
And $f(x) = - x\,\,\,\,\,if\,x \leqslant 0$
Hence, $f(x) = \dfrac{{x + 2}}{{|x + 2|}}$
If $x < - 2$
$f(x) = \dfrac{{x + 2}}{{ - (x + 2)}}$
After solving, we will get
$f(x) = - 1$
If $x > - 2$
$f(x) = \dfrac{{x + 2}}{{x + 2}}$
After solving, we will get
$f(x) = 1$
Hence, the range of the function is $\{ - 1,1\} $

Therefore, the number of elements in the range of the function is $2$

Additional Information: The modulus function, also known as the absolute value function, determines its magnitude. No matter what the number or variable, it always returns a non-negative value. The notation for a modulus function is \[y = |x|\]or $f(x) = |x|$.

Note: Students should know the properties of modulus functions and should use them to solve the given problem. And They should not include $x = - 2$ if they include that point function will not be defined.