Consider the following functions
f(x)= (7x+8)/(x+3) and g(x)= (3x-8)/(7-x)
(a) Find g(g(x))
(b) Find g(f(x))
(c) Determine whether the functions f and g are inverses of each other.
a)
g(g(x))
= g((3x-8)/(7-x)
= [3(3x-8)/(7-x) - 8]/( 7 - (3x-8)/(7-x) )
= [ (9x-24-56+8x)/(7-x)] / (49-7x-3x+8)/(7-x)]
= (17x - 80)/(57 - 10x)
b) do this in the same way I just showed you
c) let's take the inverse of f(x)
let y = (7x+8)/(x+3)
for the inverse, interchange the x and y
x = (7y+8)/(y+3)
xy + 3x = 7y + 8
xy - 7y = 8-3x
y(x-7) = 8-3x
y = (8-3x)/(x-7) = (3x - 8)/(7 - x) = g(x)
yes , they are inverses of each other.
(a) To find g(g(x)), we substitute the expression for g(x) into g(x).
g(g(x)) = g((3x-8)/(7-x))
Next, substitute (3x-8)/(7-x) into g(x) to get the expression for g(g(x)):
g(g(x)) = ((3(3x-8))/(7-(3x-8))) = ((9x-24)/(15-x))
So, g(g(x)) = (9x-24)/(15-x).
(b) To find g(f(x)), we substitute the expression for f(x) into g(x).
g(f(x)) = g((7x+8)/(x+3))
Next, substitute (7x+8)/(x+3) into g(x) to get the expression for g(f(x)):
g(f(x)) = ((3(7x+8))/(7-(7x+8))) = ((21x+24)/(-15x-49))
So, g(f(x)) = (21x+24)/(-15x-49).
(c) To determine whether the functions f and g are inverses of each other, we need to check if f(g(x)) = x and if g(f(x)) = x.
For f(g(x)):
f(g(x)) = f((3x-8)/(7-x))
Substituting (3x-8)/(7-x) into f(x):
f(g(x)) = ((7((3x-8)/(7-x))) + 8)/(((3x-8)/(7-x)) + 3)
After simplification, we get:
f(g(x)) = (21x-49)/(15-2x)
Since f(g(x)) does not equal x, f(x) and g(x) are not inverse functions.
For g(f(x)):
g(f(x)) = g((7x+8)/(x+3))
Substituting (7x+8)/(x+3) into g(x):
g(f(x)) = ((3((7x+8)/(x+3))) - 8)/(-(7x+8)/(x+3))
After simplification, we get:
g(f(x)) = (21x+24)/(-15x-49)
Since g(f(x)) does not equal x, g(x) and f(x) are not inverse functions.
Therefore, the functions f and g are not inverses of each other.
To find the value of g(g(x)), we substitute g(x) into g(g(x)):
g(g(x)) = g(3x - 8 / (7 - x))
Now, let's evaluate g(x):
g(x) = (3x - 8) / (7 - x)
Substituting g(x) into the initial expression, we get:
g(g(x)) = g(3x - 8 / (7 - x))
= [(3(3x - 8) / (7 - (3x - 8))] / (7 - (3x - 8))
= [(9x - 24) / (-x + 15)] / (-3x + 15)
Simplifying the expression further:
g(g(x)) = (9x - 24) / (-x + 15) * 1 / (-3x + 15)
= (9x - 24) / [(-x + 15)(-3x + 15)]
Now, let's move on to finding g(f(x)):
f(x) = (7x + 8) / (x + 3)
We substitute f(x) into g(x):
g(f(x)) = g[(7x + 8) / (x + 3)]
= [(3(7x + 8)) / (7 - (7x + 8))] / (7 - [(7x + 8)])
= [(21x + 24) / (-x + 15)] / (7 - 7x - 8)
= [(21x + 24) / (-x + 15)] / (-7x - 1)
Simplifying the expression further:
g(f(x)) = (21x + 24) / (-x + 15) * 1 / (-7x - 1)
= (21x + 24) / [(-x + 15)(-7x - 1)]
Lastly, let's determine whether the functions f and g are inverses of each other:
To find the inverse of f, we swap x and y in the equation and solve for y:
x = (7y + 8) / (y + 3)
x(y + 3) = 7y + 8
xy + 3x = 7y + 8
xy - 7y = 8 - 3x
y(x - 7) = 8 - 3x
y = (8 - 3x) / (x - 7)
Comparing y = (8 - 3x) / (x - 7) with g(x) = (3x - 8) / (7 - x), we can see that the functions f and g are not the same. Therefore, the functions f and g are not inverses of each other.