Please I need your help to solve and understand these questions

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

(1) Assuming you mean f(y) = 4 / (y-2) and g(x) = 5/(3x-1).

Then f comp g = 4/ [( 5/(3x-1) ) - 2] =

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

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

= (2/3) (3x-1)/ (1-x)

the domain is then all reals except 1, or (-infinity,1) U (1, infinity)

Otherwise, the denominator (3x-1) will go up to the numerator and the domain becomes all reals

Please use parenthesis to clarify what terms are in the denominator and which are not.

(2) Judging by the given data points, the function is y=4 + sqrt(x-2)

First to find the inverse, you swap x and y: x = 4 + sqrt(y-2)

Then you solve for x:

x-4 = sqrt(y-2)

(x-4)^2 = y-2

THe inverse is y = (x-4)^2 + 2

Note that f(11) = 7 and f_inverse(7) = 11

TO use desmos, when you first log on to that desmos web page, there is a red button

that says START GRAPHING. You click that button. Then the window divides into two parts:

the left window pane contains text boxes where you type the function.

For example, input hte function as y = (x+4)^2 + 2

The function will graph AS YOU TYPE IT

Each function you input shall be graphed in different colors. You can change the colors.