Any composition and inversion expert out there?

Question

I need help to finish the following exercise (I was able to finish 2 by myself but not the other ones. A step by step explanation would be highly appreciated!):

1. What can be said about the domain of the function  where  and ? Express it in terms of a union of intervals of real numbers. Go to www.desmos.com/calculator and obtain the graph of , , and . 

2. Find the inverse of the function .

3. State the domains and ranges of both the function and the inverse function in terms of intervals of real numbers. Go to www.desmos.com/calculator and obtain the graph of , its inverse, and  in the same system of axes. About what pair (a, a) are (11, 7) and (7, 11) reflected about?

Answer 1

Solution for 2).

The standard process for finding inverse of a function is as follows:

A. In the stated function we change f(x) with y.

y = 4 + √(x - 2).

B. We swap x and y in the above equation.

x = 4 + √(y - 2).

C. We solve the above equation for y (steps of the solution omitted).

y = (x - 4)² +2

Part 3. (1) Observing A above, we can say, that the domain of the function f(x) is all values of x, for which the radical has real value: x >= 2. The range of the function is all values which f(x) receives. For this function, y >=4.

(2) For the inverse, the domain and range of the inverse swap places: the domain of f becomes the range of f^-1 and vise-versa.

If you have any questions, ask question.

Note: the function f(x) is one-to-one, and when we found f^-1, it was not. This is why we must restrict the domain of the function f^-1 to only one of the parts of the parabola.

Answer 2

For (1), what you want to do is recognize that f ο g is a function of x. So, its domain will be all permissible value of x. Firstly, note that x = 1/3 is not permissible for input into the function g. So, x not equal to 1/3. Now, let's look at the composition.

f ο g (x) = 4 / (g(x)-2) = 4 / (5/(3x-1)-2) = 4 / ((5 - 2(3x-1))/(3x-1)) = 4 / ((7-6x)/(3x-1))

= (12x - 4)/(7 - 6x).

So, the domain is all x not equal to 7/6 (nor x = 1/3). So, the domain is (-infinity,1/3) union (1/3,7/6) union (7/6,infinity).

For (3), I assume the function they are referring to is the function in (2). If you found f inverse, you should be able to do this problem.