A. To find the inverse of f(x), we need to solve for x in terms of f(x).
Let y = f(x). Then we have:
y = 3/(7x + 1)
Multiplying both sides by 7x + 1, we get:
y(7x + 1) = 3
Expanding the left side, we get:
7xy + y = 3
Subtracting y from both sides, we get:
7xy = 3 - y
Dividing both sides by 7y, we get:
x = (3 - y)/(7y)
Therefore, the inverse of f(x) is:
f^-1(x) = (3 - x)/(7x)
Explanation: To find the inverse function, we set y = f(x) and solve for x. By isolating x on one side of the equation in terms of y, we obtain the inverse function.
B. Let g(x) = (3 - x)/(7x). We need to verify that (f(g(x))) = x for all values of x in the domain of f(x) and g(x).
We have:
f(g(x)) = f((3 - x)/(7x)) = 3/(7((3 - x)/(7x)) + 1) = 3/(3 - x + 7x)/(7x) = 3/(3 + 6x)/(7x) = (21x)/(3 + 6x) = 7x/(1 + 2x)
To show that (f(g(x))) = x, we need to show that:
7x/(1 + 2x) = x
Multiplying both sides by (1 + 2x), we get:
7x = x(1 + 2x)
Expanding the right side, we get:
7x = x + 2x^2
Subtracting 7x from both sides, we get:
0 = x + 2x^2 - 7x
Simplifying, we get:
2x^2 - 6x = 0
Dividing both sides by 2x, we get:
x - 3 = 0
Therefore, x = 3.
Since we have shown that (f(g(x))) = x for x = 3 (which is in the domain of both f(x) and g(x)), we can conclude that g(x) is the inverse of f(x).
C. We cannot answer part C of the question because no function is given. The table provided only shows a set of input-output pairs, but we need the function itself to perform any further analysis.
The domain and range of the inverse function f^-1(x) = (3 - x)/(7x) depend on the domain and range of the original function f(x) = 3/(7x + 1).
The domain of f(x) is all real numbers except -1/7 (since division by zero is undefined). Therefore, the range of f^-1(x) is all real numbers except 0, since division by zero is also undefined.
To find the domain of f^-1(x), we can consider the denominator of the expression for f^-1(x), which is 7x. Since division by zero is undefined, we need to exclude any value of x that makes the denominator equal to zero. Therefore, the domain of f^-1(x) is all real numbers except 0.
In summary:
Domain of f^-1(x): All real numbers except 0
Range of f^-1(x): All real numbers except 0