# What is the domain and range of f(x)=4log(x+2)-3?

Jul 10, 2018

Domain: $\left(- 2 , \infty\right)$
Range: $\left(- \infty , \infty\right)$

#### Explanation:

Given: $f \left(x\right) = 4 \log \left(x + 2\right) - 3$

It's helpful to understand the parent function: $\text{ } y = \log \left(x\right)$

Analytically, the domain is limited by the $\log$ function since a $\log$ function by definition is required to be $> 0$:

$x > 0$

The range can be any value of $y$, since the
log(.0000000001) -> -oo; " "log(100000000000) ->oo

$\text{Graph": f(x) = log(x);" }$ Domain: $\left(0 , \infty\right)$, Range: $\left(- \infty , \infty\right)$

graph{log(x) [-2, 15, -5, 5]}

The given function has a horizontal shift $2$ to the left and $3$ down. It also has a horizontal stretch of $4$.

Graph of $f \left(x\right) = 4 \log \left(x + 2\right) - 3$:

Analytically, the domain is limited by the $\log$ function since a $\log$ function by definition is required to be $> 0$:

$x + 2 > 0 \implies x > - 2$

Domain: (-2, oo); " Range: "(-oo, oo)#

graph{4 log(x+2) - 3 [-5, 15, -10, 5]}