Question

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

Answer 1

Domain: `(-2, oo)`
Range: `(-oo, oo)`

Explanation:

Given: `f(x) = 4 log(x+2) - 3`

It's helpful to understand the parent function: `" "y = log(x)`

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`

`"Graph": f(x) = log(x);" "` Domain: `(0, oo)`, Range: `(-oo, oo)`

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(x) = 4 log(x+2) - 3`:

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

`x + 2 > 0 => x > -2`

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

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