Question

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

Answer 1

Undefined at `x=4`

`{x: -oo < x < oo," " x !=4}`

Explanation:

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

Set `2x-8=0 => 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 `->` 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 `->+oo and -oo`

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

Thus as `x` tends towards positive infinity we have:
`lim_(x->+oo) (x+7)/(2x-8)->k=x/(2x)=1/2`

As `x` tends towards negative infinity we have:
`lim_(x->-oo) (x+7)/(2x-8)->-k=-x/(2x)=-1/2`

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

In set notation we have:

`{x: -oo < x < oo," " x !=4}`