Please I need help solving this question

Question

1. What can be said about the domain of the function f o g where

f (y) = 4 / y – 2 and g(x) = 5 / 3x – 1 ?

Express it in terms of a union of intervals of real numbers.

Go to desmos/calculator and obtain the graph of f , g, and f o g. (Please explain how to do the graphs)

2.   Find the inverse of the function . f(x) = 4 + √ x – 2

State the domains and ranges of both the function and the inverse function in terms of intervals of real numbers.

Go to desmos/calculator and 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?

You don't have to plot the graphs, just explain how I will go about with the graphs (with functions)

Answer 1

Here's how to do composite functions using Desmos:

1. Enter the first function, f(y) = ⁴/_y-2
2. Enter the second function g(x) = ⁵/_3x-1
3. For the composite f ο g, enter f(g(x))

For the actual calculation (this may be hard to follow because of all the fractions)

1. f(g(x)) = 4/(5/3x-1)-2

= 4/[(5/3x-1) - 2(3x-1)/(3x-1)]

= 4/[(-6x+7)/(3x-1)]

= 4(3x-1)/(-6x+7)

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

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

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

I'm going to assume that the function is f(x) = 4 + √(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 + √(f(x) -2)

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

x = 4 + √(f(x) -2)

x-4 = √(f(x) -2)

(x-4)² = f(x) -2

x² - 8x + 16 = f(x) -2

x² - 8x + 18 = f(x)

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

f^-1(x) = x² - 8x + 18

If the composite functions both evaluate to x, they are inverses of each other. In other words, f(x) ο f^-1(x) = x and f^-1(x) ο f(x) = x. I checked and they do both evaluate to x.