Probably easier than I think f(xn)=nf(x)?

  • Thread starter geoman
  • Start date
In summary, the problem is to prove that for each positive integer n and each real number x, the function f(nx) = nf(x). Using mathematical induction, we can show that f((n+1)x) = f(nx + x) = f(nx) + f(x) = nf(x) + f(x) = (n+1)f(x), which proves the desired statement.
  • #1
geoman
8
0
1. show that for each positive integer n and each real number x, f(nx)=nf(x).

Homework Equations


f(x) is an additive function (f(x+y)=f(x)+f(x))

The Attempt at a Solution


Well I'm thinking that I can just use mathematical induction to show that:
1) f((1)x)=1f(x)
2)f(nx)=nf(x)=f[tex]_{}1[/tex](x)+f[tex]_{}2[/tex](x)+...+f[tex]_{}n[/tex](x) (assume this is true)
3)f((n+1)x)=(n+1)f(x)=f[tex]_{}1[/tex](x)+f[tex]_{}2[/tex](x)+...+f[tex]_{}n+1[/tex](x)
from here it would just be algebra. is this right?
note: the numbered functions above are supposed to be subscripted but i messed it up somehow. They are only supposed to number the functions as I add the function n times and then n+1 times
 
Last edited:
Physics news on Phys.org
  • #2
How you have presented this problem is confusing.
The statement you presented in section 1 is not true in general.
The statement you presented as a relevant equation is more than just relevant; it is part of the problem statement, I'm pretty sure. Also, you have a typo, and this equation should be f(x + y) = f(x) + f(y).

In your attempt at a solution, yes you should assume that f(nx) = nf(x), but how in the world do you go from nf(x) to a sum of powers of f(x)?

Look at what happens with f(x + x). What is that equal to by your original assumptions? You have a way to go before you will be able to tackle the induction step, IMO, so let's get up to speed with some simple calculations first.
 
  • #3
geoman said:
1. show that for each positive integer n and each real number x, f(nx)=nf(x).



Homework Equations


f(x) is an additive function (f(x+y)=f(x)+f(x))



The Attempt at a Solution


Well I'm thinking that I can just use mathematical induction to show that:
Yes, induction is the way to go.

1) f((1)x)=1f(x)
2)f(nx)=nf(x)=f[tex]_{}1[/tex](x)+f[tex]_{}2[/tex](x)+...+f[tex]_{}n[/tex](x) (assume this is true)
What? Where in the world did that horribly complicated right end come from? You haven't even defined "f1", "f2", etc.
Just use f(nx)= nf(x).

3)f((n+1)x)=(n+1)f(x)=f[tex]_{}1[/tex](x)+f[tex]_{}2[/tex](x)+...+f[tex]_{}n+1[/tex](x)
You can't use "f((n+1)x)= (n+1)f(x)", that's what you want to prove. You can use the fact that f((n+1)x)= f(nx+x)= what?

from here it would just be algebra. is this right?
note: the numbered functions above are supposed to be subscripted but i messed it up somehow. They are only supposed to number the functions as I add the function n times and then n+1 times
 

What is the meaning of the function f(xn)=nf(x)?

The function f(xn)=nf(x) means that for any value of x, the output of the function is equal to n multiplied by the output of the function f(x).

What does it mean for a problem to be "probably easier" than the given function?

When a problem is described as "probably easier" than a given function, it means that the problem may seem difficult at first, but upon closer examination, it is likely to be simpler than it appears.

How can I solve a problem that involves the function f(xn)=nf(x)?

To solve a problem involving the function f(xn)=nf(x), you can use the basic rules of algebra and substitute different values of x to find the corresponding outputs. It may also be helpful to graph the function to better understand its behavior.

What are some real-life applications of the function f(xn)=nf(x)?

The function f(xn)=nf(x) has many real-life applications, such as calculating compound interest, population growth, and the rate of change in a given system. It can also be used in physics and engineering to model various phenomena.

What are some common mistakes people make when dealing with the function f(xn)=nf(x)?

One common mistake when dealing with the function f(xn)=nf(x) is forgetting to apply the function to the entire expression, resulting in incorrect outputs. Another mistake is using the incorrect value of n, which can greatly affect the overall result. It is also important to pay attention to the domain and range of the function to avoid mathematical errors.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
20
Views
385
  • Calculus and Beyond Homework Help
Replies
1
Views
457
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
604
  • Calculus and Beyond Homework Help
Replies
3
Views
165
  • Calculus and Beyond Homework Help
Replies
2
Views
870
  • Calculus and Beyond Homework Help
Replies
6
Views
983
  • Calculus and Beyond Homework Help
Replies
8
Views
347
Back
Top