For a given function f(x) we define the domain restrictions as values of x that we can not use in our function. Also, for a function f(x) we define the inverse g(x) as a function such that:
g(f(x)) = x = f(g(x))
The restriction is:
x ≠ 4
The inverse is:
[tex]y = 4 + \sqrt{\frac{11}{x} }[/tex]
Here our function is:
[tex]f(x) = \frac{11}{(x - 4)^2}[/tex]
We know that we can not divide by zero, so the only restriction in this function will be the one that makes the denominator equal to zero.
(x - 4)^2 = 0
x - 4 = 0
x = 4
So the only value of x that we need to remove from the domain is x = 4.
To find the inverse we try with the general form:
[tex]g(x) = a + \sqrt{\frac{b}{x} }[/tex]
Evaluating this in our function we get:
[tex]g(f(x)) = a + \sqrt{\frac{b}{f(x)} } = a + \sqrt{\frac{b*(x - 4)^2}{11 }}\\\\g(f(x)) = a + \sqrt{\frac{b}{11 }}*(x - 4)[/tex]
Remember that the thing above must be equal to x, so we get:
[tex]g(f(x)) = a + \sqrt{\frac{b}{11 }}*(x - 4) = x\\\\{\frac{b}{11 }} = 1\\{\frac{b}{11 }}*4 - a = 0[/tex]
From the two above equations we find:
b = 11
a = 4
Thus the inverse equation is:
[tex]y = 4 + \sqrt{\frac{11}{x} }[/tex]
If you want to learn more, you can read:
https://brainly.com/question/10300045
