Question

How do you find the domain and range of `f(x) = ln(-x + 5) + 8`?

Answer 1

Domain is `(-oo,5)` and range is `(-oo,oo)`

Explanation:

As we can have logarithm of only a positive number, we must have

`-x+5>0` or `x<5`, which gives the domain.

The least possible value of `ln(-x+5)` is `-oo` and its maximum possible value is `oo`

Hence, range is `(-oo,oo)`.

Answer 2

The domain is `x in (-oo,5)`.
The range is `y in RR`

Explanation:

The function is

`f(x)=ln(-x+5)+8`

The logarith function `lnx` is defined for `x>0`

Therefore,

`-x+5>0`

`x<5`

The domain is `x in (-oo,5)`

To find the range, let

`y=ln(-x+5)+8`

`ln(5-x)=y-8`

`5-x=e^(y-8)`

`x=5-e^(y-8)`

The exponential function `e^x` is defined over `RR`

Therefore,

The range is `y in RR`

graph{ln(5-x)+8 [-38, 35.07, -15.4, 21.14]}