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

Jul 31, 2018

Domain is $\left(- \infty , 5\right)$ and range is $\left(- \infty , \infty\right)$

#### 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 $\ln \left(- x + 5\right)$ is $- \infty$ and its maximum possible value is $\infty$

Hence, range is $\left(- \infty , \infty\right)$.

Jul 31, 2018

The domain is $x \in \left(- \infty , 5\right)$.
The range is $y \in \mathbb{R}$

#### Explanation:

The function is

$f \left(x\right) = \ln \left(- x + 5\right) + 8$

The logarith function $\ln x$ is defined for $x > 0$

Therefore,

$- x + 5 > 0$

$x < 5$

The domain is $x \in \left(- \infty , 5\right)$

To find the range, let

$y = \ln \left(- x + 5\right) + 8$

$\ln \left(5 - x\right) = y - 8$

$5 - x = {e}^{y - 8}$

$x = 5 - {e}^{y - 8}$

The exponential function ${e}^{x}$ is defined over $\mathbb{R}$

Therefore,

The range is $y \in \mathbb{R}$

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