Functions

[bardothello]

Hello.
The question is,
For exercise 37, find functions f and u so that [math]F(x)=f(u(x))[/math].
37.
[math]F(x)=(x^3+2x^2-8)^4[/math]
I have a couple questions:
1) Is the capital F important? My guess is yes, but if you could elaborate a little it would be appreciated.
2) Can you show me the steps to obtaining the answer. My book gives the answers for f and u, but I cannot figure out the steps to getting the answer. In particular, it says that [math]f(u)=u^4[/math]. I have no idea why this is.

Thanks in advance

 

[Dr.Peterson]

F and f are just two different functions. Cases are distinguished in math, but there is no necessary relationship, so I wouldn't say that f means something specific in connection with F.

The answer is not absolutely unique, but there is a most-natural division. The idea is that in function F, something (u) is done first, and something else (f) is done to the result. Looking at the function, they really are doing six things (a cube, a square, a multiplication, an addition, a subtraction, a power)! But the most obvious thing is that they first evaluate the polynomial [MATH]x^3 + 2x^2 - 8[/MATH], and then they raise that to the 4th power.

So the "inner" function is [MATH]u(x) = x^3 + 2x^2 - 8[/MATH], and the "outer" function is [MATH]f(u) = u^4[/MATH].

This concept becomes very important in calculus, and there you are looking for a natural composition like that.

 

[Steven G]

Although the method I will show below completely misses the purpose of the exercise but it always works.
Let u(x) = x and f(x) = F(x). You can even reverse the functions, let f(x) = x and let u(x) = F(x). In both cases F(x) = f(u(x)) = u(f(x))

This method will get you full credit on your exam (and possibly not thrill your instructor). At some point you really should learn what the exercise really wants, especially if you will be studying calculus.