find the domain of composite function fog
f(x)=x/x-4; g(x)= -1/x
please show work
f(g) = g/(g-4)
= (-1/x) / (-1/x - 4)
= (-1)/(-1-4x)
= 1/(4x+1)
domain is all reals except x = -1/4, 0
f(4) is not defined, but g(x) = 4 when x = -1/4.
unfortunately, g(0) is not defined, so f(g(0)) is also not defined
To determine the domain of the composite function f o g, we need to consider the domains of both functions f and g.
The domain of f(x) is all real numbers except for x = 4 because division by zero is undefined. Thus, the domain of f is (-∞, 4) U (4, ∞).
The domain of g(x) is all real numbers except for x = 0 because division by zero is undefined. Thus, the domain of g is (-∞, 0) U (0, ∞).
Now, for the composite function f o g, we substitute g(x) into f(x).
(f o g)(x) = f(g(x)) = f(-1/x) = (-1/x) / (-1/x - 4)
To determine the domain of f o g, we need to consider the restrictions on x for which the function is defined.
First, we know that x ≠ 0 due to the domain of g(x).
Next, we consider the domain of f(g(x)), which is f(-1/x). Since f(x) is defined for all real numbers except for x = 4, we can substitute -1/x into f(x) as long as -1/x ≠ 4, which simplifies to x ≠ -1/4.
Therefore, the domain of f o g is (-∞, -1/4) U (-1/4, 0) U (0, 4) U (4, ∞).
To find the domain of the composite function fog, we need to consider the restrictions on both individual functions, f(x) and g(x), as well as any restrictions arising from the composition of the two functions.
Given functions:
f(x) = x / (x - 4)
g(x) = -1 / x
First, let's identify any restrictions on the individual functions:
For f(x), the denominator (x - 4) cannot be equal to zero since division by zero is undefined. Therefore, x ≠ 4.
For g(x), the denominator (x) cannot be equal to zero since division by zero is undefined. Therefore, x ≠ 0.
Now, let's consider the composition fog(x) = f(g(x)). To find the domain of fog(x), we need to determine if there are any additional restrictions due to the composition.
f(g(x)) = f(-1 / x) = (-1 / x) / ((-1 / x) - 4)
To find the domain, we need to identify any values of x that would result in division by zero or undefined expressions. In this case, to avoid division by zero issues, we need to make sure the denominator of the composition function is not equal to zero:
(-1 / x) - 4 ≠ 0
To solve this equation, multiply both sides by x:
-1 - 4x ≠ 0
Rearrange the inequality:
-4x - 1 ≠ 0
Add 1 to both sides:
-4x ≠ 1
Divide both sides by -4 (remembering to switch the inequality sign when dividing by a negative number):
x ≠ -1/4
So, the domain of the composite function fog is all real numbers except x = 0, x = 4, and x = -1/4.