How Do You Find the Inverse of the Composite Function fh(x)?

[ghostbuster25]

Just want to check that i am doing this question correctly.

f(x) = 2x+5 h(x) = 1/x , x \neq0

Find the inverse of fh(x)

So first i found the function fh(x)

2*1/x+5

then let y = 2*1/x+5 , x \neq0


now this is the bit i can't rememeber how to do, when i try and make x the subject do i need to multiply the 2 on the RHS as well as the y on the LHS?

if i multiply the 2 then i end up with f-1(x)=2x+5/x
If i don't i end up with f-1(x) = x+7

 

[cyby]

First of all, you do mean f(h(x)), and not fh(x), which to me looks like f(x)*h(x). Under that assumption...

f(h(x)) = 2/x + 5.

So, you have y = 2/x + 5..

To find the inverse, you usually just switch x and y, and solve appropriately.

I'll start with the first step: x = 2/y + 5

Can you carry it through from here?

 

[ghostbuster25]

I can now thanks :)

I see what you have done but I am not sure why 1/x becomes 2/x in this circumstance. I am just trying to understand the mechanics behind it so i can be fully aware. If h(x) was 1/x + 5 would it still be 2/x +5 or 2/x + 10?

your correct in your assumption, my teacher is poor and makes us write it fh(x) instead of f(h(x))

 

[cyby]

Whereever you saw x, you needed to replace with 1/x. So all you really have is instead of 2*x + 5, you have 2*(1/x) + 5.

If h(x) = 1/x + 5, and f(x) = 2x + 5 then you will actually have f(h(x)) = 2(1/x + 5) + 5 = 2/x + 15.