Consider the functions


f(x)=2x+8/x+6; g(x)=6x-8/2-x
Find f(g(x))
Find g(f(x))
Deterrmine if f and g are inverses of each other.

a. What is f(g(x))?

Give any values of x that need to be excluded from f(g(x))

x=?

b. what is g(f(x))?

ive any values of x that neeedd to be excluded from g(f(x))

x=?

c. Are the functions f & g inverses of each other ?

**Please show work**

Surely you mean

f(x) = (2x+8)/(x+6) and g(x) = (6x-8)/(2-x)

f(g(x)) = (2(6x-8)/(2-x) + 8)/((6x-8)/(2-x) + 6)
multiply by (2-x)/(2-x)
= (2(6x+8) + 8(2-x))/(6x-8 + 6(2-x))
= (12x-16 + 16 - 8x)/(6x-8 + 12 - 6x)
= 4x/4
= x

check:
let x=4
g(x) = 16/-2 = -8
then f(g(4))
= f(-8) = (-16+8)/(-8+6) = -8/-2 = 4

I will leave it up to you to find g(f(x))
but you should also get x as a final answer

If f(x) and g(x) are any functions so that
f(g(x)) = x, then the two functions are inverses of each other.
We have that situations, so they are inverses of each other

or .... the long way ... actually find the inverse of one of them
let y = (2x+8)/(x+6)
inverse is
x = (2y + 8)/(y+6)
xy + 6x = 2y + 8
xy - 2y = 8 - 6x
y(x-2) = (8-6x)
y = (8-6x)/(x-2) = (6x-8)/(2-x) which is the other function.

To evaluate the composition of functions, we need to substitute the inner function into the outer function.

a. To find f(g(x)), we substitute g(x) into f(x):

f(g(x)) = 2(g(x)) + 8 / (g(x)) + 6

Substituting g(x) = (6x - 8) / (2 - x) into the equation:

f(g(x)) = 2((6x - 8) / (2 - x)) + 8 / ((6x - 8) / (2 - x)) + 6

To simplify, we find the LCD (least common denominator) for the fractions:

f(g(x)) = (2(6x - 8)(2 - x) + 8(2 - x)) / ((6x - 8) + 6(2 - x))

Next, we simplify the numerator by expanding and combining like terms:

f(g(x)) = (12x - 16 - 2x^2 + 4x + 16 - 8) / (6x - 8 + 12 - 6x)

f(g(x)) = (-2x^2 + 16x + 16) / 4

Simplifying further:

f(g(x)) = -x^2 + 8x + 4

To determine the values of x that need to be excluded, we look for any values that make the denominator of f(g(x)) equal to zero. In this case, the denominator is 4, and division by zero is not defined. Therefore, no values of x need to be excluded from f(g(x)).

b. To find g(f(x)), we substitute f(x) into g(x):

g(f(x)) = 6(f(x)) - 8 / 2 - (f(x))

Substituting f(x) = (2x + 8) / (x + 6) into the equation:

g(f(x)) = 6((2x + 8) / (x + 6)) - 8 / (2 - ((2x + 8) / (x + 6)))

To simplify, we find the LCD (least common denominator) for the fractions:

g(f(x)) = (6(2x + 8)(x + 6) - 8(x + 6)) / (2(x + 6) - (2x + 8))

Next, we simplify the numerator by expanding and combining like terms:

g(f(x)) = (12x^2 + 96x + 144 - 8x - 48) / (2x + 12 - 2x - 8)

g(f(x)) = (12x^2 + 88x + 96) / 4

Simplifying further:

g(f(x)) = 3x^2 + 22x + 24

Similar to part a, we need to exclude any values of x that make the denominator of g(f(x)) equal to zero. In this case, the denominator is 4, and division by zero is not defined. Therefore, no values of x need to be excluded from g(f(x)).

c. To check if f and g are inverses of each other, we need to verify if f(g(x)) = x and g(f(x)) = x.

We found that f(g(x)) = -x^2 + 8x + 4, and g(f(x)) = 3x^2 + 22x + 24.

Since f(g(x)) is not equal to x and g(f(x)) is not equal to x, the functions f and g are not inverses of each other.