Question

Given r(x) = 11/ (x - 42)

Answer 1

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:

$y = 4 + \sqrt{\frac{11}{x} }$

Here our function is:

$f(x) = \frac{11}{(x - 4)^2}$

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:

$g(x) = a + \sqrt{\frac{b}{x} }$

Evaluating this in our function we get:

$g(f(x)) = a + \sqrt{\frac{b}{f(x)} } = a + \sqrt{\frac{b*(x - 4)^2}{11 }}\\\\g(f(x)) = a + \sqrt{\frac{b}{11 }}*(x - 4)$

Remember that the thing above must be equal to x, so we get:

$g(f(x)) = a + \sqrt{\frac{b}{11 }}*(x - 4) = x\\\\{\frac{b}{11 }} = 1\\{\frac{b}{11 }}*4 - a = 0$

From the two above equations we find:

b = 11

a = 4

Thus the inverse equation is:

$y = 4 + \sqrt{\frac{11}{x} }$

If you want to learn more, you can read:

brainly.com/question/10300045