SOLUTION: f(x)= (-2 over x+5) -1 g(x)= x-2 determine the composite function f(g(x)), and then find the inverse of that function. state domain and range. then graph the composite function

Algebra ->  Rational-functions -> SOLUTION: f(x)= (-2 over x+5) -1 g(x)= x-2 determine the composite function f(g(x)), and then find the inverse of that function. state domain and range. then graph the composite function      Log On


   

Question 118159: f(x)= (-2 over x+5) -1
g(x)= x-2
determine the composite function f(g(x)), and then find the inverse of that function. state domain and range. then graph the composite function and its inverse.
PLEASE HELP ME THIS WILL PRACTICALLY CHANGE MY LIFE IF YOU CAN HELP ME.
thank you for helping me, you are the greatest!!!!
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Change your life?
As long as there's no pressure.
f%28x%29=%28-2%29%2F%28x%2B5%29-1
g%28x%29=x-2
To find the composite f(g(x)), substitute (x-2) where there is an x in f(x).
f%28g%28x%29%29=%28-2%29%2F%28%28x-2%29%2B5%29-1
f%28g%28x%29%29=%28-2%29%2F%28x%2B3%29-1
The domain of f(g(x)) is all points except x=-3, since the denominator goes to zero.
For the range of f(x), as x gets large, either positive or negative, the term (-2)/(x+3) goes to zero and
f%28x%29-%3E-1
Here is the graph of f(g(x)).
+graph%28+300%2C+300%2C+-20%2C+20%2C+-5%2C+5%2C%28-2%29%2F%28x%2B3%29-1%29+
That was f(g(x)).
Now for the inverse.
To find the inverse of
y=%28-2%29%2F%28x%2B3%29-1
Reverse x and y and solve for y.
y=%28-2%29%2F%28x%2B3%29-1 Original equation of f(g(x)).
highlight%28x%29=%28-2%29%2F%28highlight%28y%29%2B3%29-1 Interchange x and y.
x%28y%2B3%29=-2-%28y%2B3%29 Multiply both sides by (y+3).
xy%2B3x=-2-y-3 Distributive property.
xy%2B3x=-y-5 Simplify.
xy%2By=-3x-5 Group like terms (y on left, x on right).
y%28x%2B1%29=-%283x%2B5%29Distributive property.
y=-%283x%2B5%29%2F%28x%2B1%29Divide by (x+1).
There it is.
f%28g%28x%29%29%5E-1=-%283x%2B5%29%2F%28x%2B1%29
Domain of the inverse function is all numbers except x=-1.
Think about the range of f(g(x)) and the domain of the inverse and you'll see why.
Hint : You never reach -1 in f(g(x)) except at infinity
Range of the inverse function: as x gets large, the inverse function looks like,
y=-%283x%2Fx%29=-%283cross%28x%29%2Fcross%28x%29%29=-3
Again think about the range of the inverse and the domain of the f(g(x)) for reasons why.
Here's the graph of the inverse.
+graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C-%283x%2B5%29%2F%28x%2B1%29%29