Is the Function f(x) Odd or Even? Proving with Real Numbers

In summary, the conversation discusses solving for whether a function that satisfies the statements f(x+1)=f(x)+1 and f(x^2) =(f(x))^2 is odd or even. The attempt at a solution involves using the given equations to find specific function values and then discussing how to prove that the function is odd. Continuity is also mentioned as a possible factor in determining the function's properties.
  • #1
rattanjot14
16
0

Homework Statement



1) f(x+1)=f(x)+1
2) f(x^2) =(f(x))^2
let a function real to real satisfy the above statements then prove whether the fuction is odd or even.

Homework Equations


The Attempt at a Solution


using the 2) we get 1) f(0) = 0,1
2) f(1) = 0,1
putting x = 0 in the 1st equation we get f(0) = 0 and f(1) = 1. from this we can prove f(-1) = -1 and for integers we get f(-x) = -f(x). But how to prove for real
 
Physics news on Phys.org
  • #2
Please do not post threads in multiple forums.

Did you consider ##f(\sqrt{n}^2)## for integers n?
What about f(1/2) and f(-1/2)?

I am not sure if it is possible to construct the whole function in that way. Based on your function values, it is trivial to show that the function cannot be even, but that is not sufficient to show that it is odd. Continuity would be nice to have.
 
  • #3
We can say it is not an even function ..then how to prove that it is an odd function.
 
  • #4
What can you say about |f(x)| compared with|f(-x)| on the one hand, and f(|x|) compared with f(-|x|) on the other?
 

FAQ: Is the Function f(x) Odd or Even? Proving with Real Numbers

What is an even function?

An even function is a mathematical function that satisfies the property f(x) = f(-x) for all values of x in its domain. This means that the graph of an even function is symmetric with respect to the y-axis.

What is an odd function?

An odd function is a mathematical function that satisfies the property f(x) = -f(-x) for all values of x in its domain. This means that the graph of an odd function is symmetric with respect to the origin.

How can you tell if a function is even or odd?

To determine if a function is even or odd, you can use the symmetry properties mentioned above. If the function is symmetric with respect to the y-axis, it is even. If it is symmetric with respect to the origin, it is odd. Another way to tell is by looking at the powers of x in the function. Even functions have only even powers of x, while odd functions have only odd powers of x.

What is the difference between even and odd functions?

The main difference between even and odd functions is their symmetry properties. Even functions are symmetric with respect to the y-axis, while odd functions are symmetric with respect to the origin. Additionally, even functions have only even powers of x, while odd functions have only odd powers of x.

Why are even and odd functions important?

Even and odd functions are important in mathematics because they have many useful properties and applications. For example, even functions are often used to model symmetrical systems, while odd functions are used to model anti-symmetrical systems. They also play a key role in Fourier series and other advanced mathematical concepts.

Similar threads

Is ##f(x)=2^{x}-1## considered an exponential function?
Replies
12
Views
2K
To find the boundedness of a given function
Replies
14
Views
1K
State the domain and range for a given function
Replies
7
Views
1K
To find the range of a given ##\sin## function
Replies
15
Views
1K
Find the range of the function ##f(x) = \frac{e^{2x}-e^x+1} {e^{2x}+e^x+1}##
Replies
8
Views
1K
Solve the problem involving the cubic function
Replies
12
Views
1K
If p is even and q is odd, then ##\binom{p}{q}## is even
Replies
19
Views
1K
Prove that ## g(x)=f(x)\log {x}-\int_{2}^{x}t^{-1}f(t)dt ##.
Replies
7
Views
1K
Prove that ## li(x)\sim \frac{x}{\log x} ##.
Replies
3
Views
1K
Determine whether a logarithmic function is odd or even
Replies
11
Views
5K

Hot Threads

Recent Insights

Back
Top