How do you find the range of $f(x)=\frac{x+3}{|x-2|}$?

Question

I know how to find the Range when the Modulus/ Absolute-Value function is in the Numerator. But how do I solve it when a modulus function is in the denominator? Can you please explain it with this example: $$f(x)= \frac{x+3}{|x-2|}$$

When $x>2$, $$f(x)= \frac{x+3}{x-2},$$ whose range is $\mathbb{R}-\{1\}$. When $x<2$, $$f(x)= \frac{x+3}{-x+2},$$ whose range is $\mathbb{R}-\{-1\}$. So the Final Range set should be $\mathbb{R} - \{-1,1\}$. But plotting the function in Desmos Graph Shows that Range is $(-1, \infty)$. That's where I'm getting confused.

Answer 1

$$f(x)= \frac{x+3}{|x-2|}$$

When $x\ge2$, $$g(x)= \frac{x+3}{x-2},$$ whose range is $\mathbb{R}-\{1\}$.

Yes, but what you want is not $g$ but its restriction $g|_{[2,\infty)},$ which has range $(1,\infty).$

When $x<2$, $$h(x)= \frac{x+3}{-x+2},$$ whose range is $\mathbb{R}-\{-1\}$.

Similarly, you want $h|_{(-\infty,2)},$ which has range $(-1,\infty).$

But plotting the function in Desmos Graph Shows that Range is $(-1, \infty).$

Yes, this is the union of the two ranges above.