SOLUTION: can you tell me i have been trying to get the answer and I just can't get it . This two function with the works given f(x)= 1/x-2 G(x)=4/x. Find (fog)(x) and using the same funct

Algebra ->  Rational-functions -> SOLUTION: can you tell me i have been trying to get the answer and I just can't get it . This two function with the works given f(x)= 1/x-2 G(x)=4/x. Find (fog)(x) and using the same funct      Log On


   

Question 293389: can you tell me i have been trying to get the answer and I just can't get it . This two function with the works given f(x)= 1/x-2 G(x)=4/x. Find (fog)(x) and using the same function find (gof)(x). thank you in advance
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
f(x) = 1%2F%28x-2%29

g(x) = 4%2Fx

Find (fog)(x)

Find (gof)(x)

First we'll do (fog)(x):

(fog)(x) = f(g(x))

g(x) = %284%2Fx%29

f(x) = %281%2F%28x-2%29%29

f(g(x)) becomes f(%284%2Fx%29)

You are replacing x with g(x) in f(x) which means you are replacing x with 4%2Fx in f(x).

Your equation f(x) becomes f(%284%2Fx%29)

Since f(x) = 1%2F%28x-2%29 and you are replacing x with (4/x), then your equation becomes:

f(%284%2Fx%29) = 1%2F%28%284%2Fx%29-2%29

If you multiply numerator and denominator of this expression by x, then you get:

f(%284%2Fx%29) = x+%2F+%28x%2A%28%284%2Fx%29+-+2%29%29 which becomes:

f(%284%2Fx%29) = x+%2F+%284-2x%29

To test this out, pick a number for x.

Let x = 5.

f(g(4%2Fx)) = x%2F%284-2x%29 = 4%2F%284-8%29 = 4%2F%28-4%29 = -1

Solve separately to confirm the answer is good.

Solve for g(x) = 4%2Fx to get g(4) = 4%2F4 = 1

You now have g(x) = 1

Solve for f(g(x)) = f(1) = 1%2F%28x-2%29 = 1%2F%281-2%29 = 1%2F%28-1%29 = -1

We get a match again, so the function f(g(x)) was translated correctly.

Next we'll do (gof)(x).

That's equivalent to g(f(x)).

f(x) = 1%2F%28x-2%29

g(x)= 4%2Fx

g(f(x)) is therefore equal to 4%2F%281%2F%28x-2%29%29

Multiply numerator and denominator of this equation by (x-2) to get:

g(f(x)) = %284%2A%28x-2%29%29%2F1 = 4x-8

To confirm we'll do g(x) separately, and then do f(g(x)) from the answer, and we'll do f(g(x)) using the combined formula to see if they match.

We'll choose x = 5 again.

g(f(x)) = 4x-8 = 4*5 - 8 = 20 - 8 = 12

Solve for f(x) separately to get f(x) = 1%2F%28x-2%29 = 1%2F%285-2%29 = 1%2F3

solve for g(f(x)) = g(1%2F3) to get g(1%2F3) = 4%2F%281%2F3%29 = 4%2A3 = 12.

The answers match.

Those should be your answers:

f(g(x)) = x+%2F+%284-2x%29

g(f(x)) = 4x-8