If #f(x) = 1/x#, and #g(x) = sqrt(x-2)#, how do you find the domain and range of (f o g)(x)?

Question

Cora Alexander · Accepted Answer

D:#(2, +oo)#
R:#(0, +oo)# Given : #f(x)=1/x# and #g(x)=sqrt(x-2)# #f (g(x))=1/g(x)=1/sqrt(x-2)# Range should be #x>2# or #(2, oo)#. It cannot include 2 because that will make the function undefine because of #1/sqrt(0-0)# God bless you....

Sadie Alexander · Answer

undefined To find the domain and range of the composition (f o g)(x), first find the domain of g(x), then determine the range of g(x) and use it to find the domain of f(x) when g(x) is its input. Once you find the domain of f o g, determine the range of f(x) and use it to find the range of f o g(x).

1. Domain of g(x):
Since g(x) involves a square root, the expression inside the square root (x - 2) must be greater than or equal to zero.
x - 2 ≥ 0
x ≥ 2
So, the domain of g(x) is [2, ∞).

2. Domain of f o g(x):
The domain of f(x) is all real numbers except for x = 0 since division by zero is undefined. Since g(x) is the input for f(x), the domain of f(x) when g(x) is its input would exclude any values where g(x) equals zero. However, g(x) is never zero in its domain [2, ∞). So, the domain of f o g(x) is [2, ∞).

3. Range of g(x):
Since g(x) is a square root function, its range is all non-negative real numbers. So, the range of g(x) is [0, ∞).

4. Range of f o g(x):
To find the range of f o g(x), we need to consider the range of g(x), which is [0, ∞). The function f(x) = 1/x has a range of all real numbers except for 0. Since g(x) is always non-zero in its domain [2, ∞), the composition f(g(x)) = f o g(x) will not have zero in its range.
So, the range of f o g(x) is (-∞, 0) ∪ (0, ∞).