Please help me with function decomposition? Pre-Calculus?? Very confused!

Question

1. Given the composition h(x)=f(g(x))

If h(x)=(x+3)^8 and f(x)=x^8, find the function g(x).

 

 

 

 

 

2. Given the composition h(x)=f(g(x))

If h(x)=1/x+3 and g(x)=x+3 , find the function f(x). 

 

 

I am SO confused with how to solve these! The other articles on decomposition online never provide three separate functions/variables. Please explain if you can. If not, that is okay, I just want some help if possible. All assistance is greatly appreciated!!!!

Answer 1

1. Given the composition h(x)=f(g(x))
If h(x)=(x+3)⁸ and f(x)=x⁸, find the function g(x).

 

To figure this out, h(x) = f(g(x)) means that

 

h(x) = (g(x))⁸         

 

because you plug g(x) into f(x) wherever you see an x when you take f(g(x)). But the problem also says that 

 

h(x) = (x + 3)⁸

 

Then g(x) must be the same thing as x + 3.

 

So g(x) = x + 3

 

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2. Given the composition h(x)=f(g(x))

If h(x)=1/(x+3) and g(x)=x+3 , find the function f(x).

 

h(x) = f(g(x)) and since g(x) = x + 3,

 

h(x) = f(g(x)) = f(x + 3)  which means that wherever I see an x in f(x) I put an x + 3

 

but h(x) = 1/(x + 3)

 

which means that f(x) must be 1/x in order for f(x + 3) to be 1/(x + 3)

 

so f(x) = 1/x

 

Answer 2

1. Let g(x) = x+3.  Then h(x) = f(g(x)) = f(x+3) = (x+3)⁸

 

2. Let f(x) = 1/x.  Then, h(x) = f(g(x)) = f(x+3) = 1/(x+3)