Question

Find the inverse of the function
f(x)=4+ \sqrt{x-2} .
State the domains and ranges of both the function and the inverse function in terms of intervals of real numbers.
obtain the graph of f , its inverse, and g(x)=x in the same system of axes. About what pair (a, a) are (11, 7) and (7, 11) reflected about?

Answer 1

$$Solution$$

To find the inverse of a function there are 2 steps:

1. Everywhere there's a $y, f(x), g(x),$ etc, swap it with x
2. Solve for $y$ in the rewritten function

I'm going to assume that the function is $f(x) = 4 + \sqrt{x-2}$ where the $x-2$ is under the radical sign

So the first step is to swap the $f(x)$ and $x$ . The new function would be

$x = 4 + \sqrt{f(x) -2}$

Now we'll solve for $f(x)$ in the new function

$x = 4 + \sqrt{f(x) -2}\\ x-4 = \sqrt{f(x) -2}\\ (x-4)^2 = f(x) -2\\ x^2 - 8x + 16 = f(x) -2\\ x^2 - 8x + 18 = f(x)$

So the inverse function, which we write as $f^{-1}(x)$ is

$$f^{-1}(x) = x^2 - 8x + 18$$

Domain of $f(x)$

$$x-2 \geq 0\\ \implies x \geq 2$$

Range of $f(x)$

$$[4, \infin)$$

From the graph, the points $(11, 7)$ and $(7, 11)$ reflected about point $(9,9)$.