Let f:[rr]-> [rr]

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In summary, the conversation discusses a function f:[rr]--> [rr] and its properties, including f(xy)=f(x)+f(y) and f(xn)=nf(x). The conversation also touches on finding the inverse of f and its relationship to the original function, specifically showing that for all integers n, [f-1(x)]n=f-1(nx). The conversation concludes with a recommendation to use a toolkit for solving math problems involving the inverse of a function.
  • #1
Let f:[rr]--> [rr]

It isn't a homework problem. :smile:
Let f:[rr]--> [rr] be a function such that f(xy)=f(x)+f(y) for all x,y belong to real numbers
a) Show that for all integers n, f(xn)=nf(x)
b) Suppose the inverse of f exists. Show that for all integers n, [f-1(x)]n=f-1(nx)

I know how to do part (a) but not part (b). Could someone please give me some hints/solution. I can't do problems involving "inverse of a function" unless they are very simple, like to find f-1 of f(x)=(1-x)/(1+x) where x does not equal to 1 or -1
 
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  • #2


Originally posted by KL Kam
It isn't a homework problem. :smile:
Let f:[rr]--> [rr] be a function such that f(xy)=f(x)+f(y) for all x,y belong to real numbers
a) Show that for all integers n, f(xn)=nf(x)
b) Suppose the inverse of f exists. Show that for all integers n, [f-1(x)]n=f-1(nx)

I know how to do part (a) but not part (b). Could someone please give me some hints/solution. I can't do problems involving "inverse of a function" unless they are very simple, like to find f-1 of f(x)=(1-x)/(1+x) where x does not equal to 1 or -1

a) that's the power rule formula for solving dx
b) this is antidx. remember its the opposite of dx. if your good with the power rule just do it backwards and subtract.

A good http://math.vanderbilt.edu/~pscrooke/toolkit.shtml [Broken] for you to use.
dx :wink:
 
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  • #3
inverse

Let x=f(y), or y=f-1(x)
f(yn)=nf(y)=nx
Take inverse on both sides and get
yn=(f-1(x))n=f-1(nx)
 

1. What does "f:[rr]-> [rr]" mean?

"f:[rr]-> [rr]" is a notation used in mathematics to represent a function. The first "rr" refers to the input or domain of the function, and the second "rr" refers to the output or range of the function. This notation is commonly used in calculus and other branches of mathematics.

2. What is the purpose of using "f:[rr]-> [rr]"?

The notation "f:[rr]-> [rr]" helps to clearly define the input and output of a function. It also allows for easy communication and understanding among mathematicians, scientists, and other professionals who use mathematical functions in their work.

3. Can "f:[rr]-> [rr]" be used for all types of functions?

Yes, "f:[rr]-> [rr]" can be used for all types of functions, including linear, quadratic, exponential, trigonometric, and many others. It is a general notation that can be applied to any type of function.

4. How do you read "f:[rr]-> [rr]"?

The notation "f:[rr]-> [rr]" can be read as "f of rr equals rr." This means that the function f takes an input of rr and produces an output of rr.

5. Are there any other notations for representing a function?

Yes, there are other notations that can be used to represent a function, such as "y = f(x)" or "f(x) = x^2 + 3x + 1." These notations are commonly used in algebra and calculus, but they all convey the same concept of a function mapping inputs to outputs.

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