Question

How do you find the domain and range of $\dfrac{{{x^2} - 64}}{{x - 8}}?$

Answer 1

Hint: The given question consists of fractional functions, in which both numerator and denominator are polynomials. And the domain of a polynomial is all real values but here the polynomial in the denominator should not equal to zero so find its zero and remove it from the domain of polynomials. Then simplify the fraction and put the right and left limit of the domain to find the range.

Complete step-by-step solution:
In order to find the domain and range of the given function $\dfrac{{{x^2} - 64}}{{x - 8}}$, we can see that the numerator and the denominator are both polynomials, so their individual domains will be set of real numbers, but we have been asked to find the domain of the function.
Here in the function $y = \dfrac{{{x^2} - 64}}{{x - 8}}$ we can see that it is a fractional function consisting of two polynomial functions as numerator and denominator, so the criteria of a fractional function to be defined is only that its denominator should not equals to zero,
$
   \Rightarrow x - 8 \ne 0 \\
   \Rightarrow x \ne 8 \\
 $
Therefore domain: \[R - \{ 8\} = \left( { - \infty ,\;8} \right) \cup \left( {8,\;\infty } \right)\]
From the function
$
   \Rightarrow y = \dfrac{{{x^2} - 64}}{{x - 8}} \\
   \Rightarrow y = \dfrac{{{x^2} - {8^2}}}{{x - 8}} \\
   \Rightarrow y = \dfrac{{(x + 8)(x - 8)}}{{x - 8}}\;\;\;\left[ {{\text{using}}\;{a^2} - {b^2} = (a + b)(a - b)} \right] \\
   \Rightarrow y = x + 8 \\
 $
Putting here the right and left limit of the domain, we will get
\[
   \Rightarrow y = - \infty + 8\;{\text{and}}\;y = \infty + 8 \\
   \Rightarrow y = - \infty \;{\text{and}}\;y = \infty \\
 \]
But domain does not include $8$
\[
   \Rightarrow y = 8 + 8 \\
   \Rightarrow y = 16 \\
 \]
Therefore range will not include $16$

That is, range: \[R - \{ 16\} = \left( { - \infty ,\;16} \right) \cup \left( {16,\;\infty } \right)\]

Note: When finding the range must cross check that you have excluded the values which are coming when we put the critical values in the function. You can also find the range by finding the inverse of the function and then find the respective domain of that inverse function, the domain of the inverse function will be the range of the original function.