Question

How do you find the range for $f\left( x \right) = \dfrac{{{x^2} - 4}}{{x - 2}}$?

Answer 1

In the given question, we have been asked the range of a given function. To determine the domain, we just check what values the argument of the function have. We subtract the values which cannot be substituted into the argument. To find the range, we just see what answer could the given expression possibly have and that gives us the answer.

Complete step by step solution:
The given function is $f\left( x \right) = \dfrac{{{x^2} - 4}}{{x - 2}}$.
For finding the domain, we just consider the argument, which is $\dfrac{{{x^2} - 4}}{{x - 2}}$.
Clearly, $\dfrac{1}{{x - 2}}$ can take any value as input except for $0$, as this is going to give us $\dfrac{1}{0}$, which is an indeterminate form.
Hence, the domain is R - \left\\{ 2 \right\\}.
Now, $f\left( x \right) = \dfrac{{{x^2} - 4}}{{x - 2}}$
Let us simplify the value,
$f\left( x \right) = \dfrac{{{x^2} - 4}}{{x - 2}} = \dfrac{{\left( {x + 2} \right)\left( {x - 2} \right)}}{{\left( {x - 2} \right)}} = x + 2$
Hence, the range can have any value except,
Hence, the range is $R$

Note: To find the domain – the values which can be put in the function, we just consider the argument and subtract the points where the argument yields an indeterminate form. For range, we see what values are possible in the given question. And that gives us the answer.