Find the domain of the composite function FOG
f(x)= 2/x-3; g(x)= 7/x
please show work so I can undestand
(fog)(x) = f(g(x))
f(x) = 2/(x-3), so
f(g) = 2/(g-3), but
g(x) = 7/x, so
f(g(x)) = 2/(7/x - 3) = 2x/(7-3x)
the domain is all reals except where the denominator is zero. so, all reals except x = 7/3
To find the domain of the composite function FOG, we need to consider the domains of both individual functions f(x) and g(x), and then determine the values of x for which the composite function FOG is defined.
Given:
f(x) = 2/(x-3)
g(x) = 7/x
First, let's find the domain of f(x):
The expression 2/(x-3) is defined for all real numbers except when the denominator (x-3) is equal to zero. So, we set the denominator equal to zero and solve for x:
x - 3 = 0
x = 3
Therefore, the function f(x) is undefined at x = 3. Hence, the domain of f(x) is all real numbers except x = 3.
Next, let's find the domain of g(x):
The expression 7/x is defined for all real numbers except when the denominator (x) is equal to zero. So, x = 0 is not allowed in the domain of g(x).
Now, to find the domain of the composite function FOG, we need to consider the combination of both individual domains.
Since f(x) = 2/(x-3) and g(x) = 7/x, the composite function FOG can be written as: FOG(x) = f(g(x)) = f(7/x)
To evaluate f(7/x), we need to substitute 7/x into the function f(x):
f(7/x) = 2/(7/x - 3)
Now, let's determine the values of x for which the function f(7/x) is defined:
The expression 7/x - 3 is defined for all real numbers except when the denominator (7/x - 3) is equal to zero.
Setting the denominator equal to zero:
7/x - 3 = 0
7 - 3x = 0
3x = 7
x = 7/3
Therefore, the function f(7/x) is undefined at x = 7/3. Thus, the domain of FOG is all real numbers except x = 7/3, x = 3, and x = 0.
In conclusion, the domain of the composite function FOG, with f(x) = 2/(x-3) and g(x) = 7/x, is all real numbers except x = 7/3, x = 3, and x = 0.