# What is the domain and range of f(x)= (x+7)/(2x-8)?

Jun 1, 2017

Domain: $= \left\{x | x \ne 4\right\}$
Range $= \left\{y | y \ne 0.5\right\}$

#### Explanation:

Disclaimer : My explanation may be missing some certain aspects due to the fact that I am not a professional mathematician.

You can find both the Domain and Range by graphing the function and seeing when the function is not possible. This may be a trial and error and take some time to do.
You can also try the methods below

Domain
The domain would be all the values of $x$ for which the function exists. Hence, we can write for all the values of $x$ and when $x \ne$ a certain number or numbers. The function will not exist when the denominator of the function is 0. Hence we need to find when it does equal 0 and say that the domain is when $x$ does not equal the value we find:
$2 x - 8 = 0$

∴2x=8

∴x=8/2

∴x=4

When $x = 4$, the function is not possible, as it becomes $f \left(x\right) = \frac{2 + 7}{0}$ which is undefined, hence not possible.

Range
To find the range, you can find the domain of the inverse function, to do this, rearrange the function to get x by itself. That would get quite tricky.

or

We can find the range by finding the value of y for which $x$ approaches $\infty$ (or a very big number). In this case we will get
$y = \frac{1 \left(\infty\right) + 7}{2 \left(\infty\right) - 8}$

As $\infty$ is a very big number the $+ 7$ and the $- 8$ wont change it much, Hence we can get rid of them. We are left with:
$y = \frac{1 \left(\infty\right)}{2 \left(\infty\right)}$
The $\infty$'s can cancel out, and we are left with
$y = \frac{1}{2}$
Hence the function is not possible for when $y = \frac{1}{2}$

A short way to do this is to get rid of everything except for the constants for the variables (the numbers in front of the $x$'s)
∴y=x/(2x)->1/2

Hope that's helped.

Jun 1, 2017

$x \in \mathbb{R} , x \ne 4$
$y \in \mathbb{R} , y \ne \frac{1}{2}$

#### Explanation:

$\text{y = f(x) is defined for all real values of x, except for any}$
$\text{that make the denominator equal zero}$

$\text{equating the denominator to zero and solving gives}$
$\text{the value that x cannot be}$

$\text{solve " 2x-8=0rArrx=4larrcolor(red)" excluded value}$

$\text{domain is } x \in \mathbb{R} , x \ne 4$

$\text{to find any excluded values in the range, rearrange}$
$\text{f(x) making x the subject}$

$\Rightarrow y \left(2 x - 8\right) = x + 7 \leftarrow \textcolor{b l u e}{\text{ cross-multiplying}}$

$\Rightarrow 2 x y - 8 y = x + 7$

$\Rightarrow 2 x y - x = 7 + 8 y$

$\Rightarrow x \left(2 y - 1\right) = 7 + 8 y$

$\Rightarrow x = \frac{7 + 8 y}{2 y - 1}$

$\text{the denominator cannot equal zero}$

$\text{solve " 2y-1=0rArry=1/2larrcolor(red)" excluded value}$

$\text{range is } y \in \mathbb{R} , y \ne \frac{1}{2}$