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

1 Answer
Jul 10, 2018

Answer:

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]}