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

Jul 14, 2017

Undefined at $x = 4$

$\left\{x : - \infty < x < \infty , \text{ } x \ne 4\right\}$

#### Explanation:

You are not 'allowed' to divide by 0. The proper name for this is that the function is 'undefined'. at that point.

Set $2 x - 8 = 0 \implies x = + 4$

So the function is undefined at $x = 4$. Sometimes this is referred to as a 'hole'.
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Domain and Range $\to$ letters d and r

In the alphabet d comes before r and you have to input ($x$) before you get an output ($y$).

So you consider the range as the values of the answer.

So we need to know the values of $y$ as $x$ tends to positive and negative infinity $\to + \infty \mathmr{and} - \infty$

As $x$ becomes exceptionally big then the effect of the 7 in $x + 7$ is of no importance. Likewise the effect of -8 in $2 x - 8$ becomes of no importance. My use of $\to$ means 'tends towards'

Thus as $x$ tends towards positive infinity we have:
${\lim}_{x \to + \infty} \frac{x + 7}{2 x - 8} \to k = \frac{x}{2 x} = \frac{1}{2}$

As $x$ tends towards negative infinity we have:
${\lim}_{x \to - \infty} \frac{x + 7}{2 x - 8} \to - k = - \frac{x}{2 x} = - \frac{1}{2}$

So the range is all values between negative infinity and positive infinity but excluding 4

In set notation we have:

$\left\{x : - \infty < x < \infty , \text{ } x \ne 4\right\}$