Function, f, with domain (-infinity, + infinity)

In summary, We discuss about the problem of finding the special symmetry of the function g(x) = f(x) + f(-x) and determine that it is an even function. We also explore the concepts of even and odd functions and how they relate to the original function. Different approaches to solving the problem are also mentioned.
  • #1
ajassat
55
0
I was working on the following problem from a textbook. The textbook has no answer. I have included my solution - I am not sure whether it is correct Any ideas and or solutions? (guidance)

Question:
Suppose that f is any function with domain (-infinity, +infinity)

a) Does the function g defined by g(x) = f(x) + f(-x) have any special symmetry?

My solution

A function f which satisfies:
f(-x) = -f(x) is symmetrical about the horizontal axis. This is also:
f(x) + f(-x) = 0
or we could substitute values of x giving non zero solutions for:
f(x) + f(-x) = g(x) (some other function)

Therefore g(x) = f(x) + f(-x) has symmetry about horizontal axis?

Any guidance appreciated.

Regards,
Adam
 
Mathematics news on Phys.org
  • #2
No. f(x)+f(-x) is an even function: it's symmetric about the vertical axis.
 
  • #3
Try and work backwards: take an arbitrary function [tex]g[/tex] defined on [tex]\mathbb{R}[/tex] and try to create a function [tex]f[/tex] s.t. [tex] f(x) + f(-x) = g(x)[/tex] for all x. What assumptions must be made about [tex]g[/tex] in order to do this?
 
  • #4
ajassat said:
I was working on the following problem from a textbook. The textbook has no answer. I have included my solution - I am not sure whether it is correct Any ideas and or solutions? (guidance)

Question:
Suppose that f is any function with domain (-infinity, +infinity)

a) Does the function g defined by g(x) = f(x) + f(-x) have any special symmetry?

My solution

A function f which satisfies:
f(-x) = -f(x) is symmetrical about the horizontal axis. This is also:
f(x) + f(-x) = 0
or we could substitute values of x giving non zero solutions for:
f(x) + f(-x) = g(x) (some other function)

Therefore g(x) = f(x) + f(-x) has symmetry about horizontal axis?

Any guidance appreciated.

Regards,
Adam
Just use the definition of odd and even functions for this type of problem. Without making any assumptions about f(x) (apart from the given domain) simply write g(-x) and compare it with g(x). That is,

g(x) = f(x) + f(-x) : given

So g(-x) = f(-x) + f(-(-x))
= f(-x) + f(x)
= g(x).

And g(-x) = g(x) is the definition of an even function.
 
Last edited:
  • #5
I figured it was an even function after I had posted this.

Thanks to both some_dude and uart for posting two different approaches to the problem. I understand this problem a lot better and have solved some similar now.

Thanks again,
Adam
 
  • #6
To test yourself, a new question: do the same for g(x)=f(x)-f(-x).
 
  • #7
By the way, f(x)+ f(-x) and f(x)- f(-x) are almost the "even and odd parts" of the function f(x). e(x)= (f(x)+ f(-x))/2 and o(x)= (f(x)- f(-x))/2 are even and odd functions that add to give f(x). Of course, if f(x) is an even function to begin with, o(x) will be 0 and e(x)= f(x). If f(x) is an odd function, e(x)= 0 and o(x)= f(x).

The exponential function, ex, is neither even nor odd. Its even and odd parts are [itex]\frac{e^x+ e^{-x}}{2}= cosh(x)[/itex] and [itex]\frac{e^x- e^{-x}}{2}= sinh(x)[/itex].
 

What is a function with a domain of (-infinity, +infinity)?

A function with a domain of (-infinity, +infinity) is a mathematical relationship between two variables where the domain (or input) can take on any real number from negative infinity to positive infinity. This means that there are no restrictions on the input values for the function.

What is the range of a function with a domain of (-infinity, +infinity)?

The range of a function with a domain of (-infinity, +infinity) can vary depending on the specific function. However, the range will always be a subset of the real numbers, as the output of the function must be a real number.

Can a function with a domain of (-infinity, +infinity) be continuous?

Yes, a function with a domain of (-infinity, +infinity) can be continuous. This means that there are no breaks or gaps in the graph of the function, and it can be drawn without lifting the pen from the paper.

Is it possible for a function with a domain of (-infinity, +infinity) to have a vertical asymptote?

No, a function with a domain of (-infinity, +infinity) cannot have a vertical asymptote. This is because there are no restrictions on the input values for the function, so the graph will never approach a specific x-value infinitely.

What are some real-life examples of functions with a domain of (-infinity, +infinity)?

Some real-life examples of functions with a domain of (-infinity, +infinity) include distance traveled over time, temperature over time, and the growth of a population over time. These functions have a continuous domain and can take on any real value for the input variable.

Similar threads

Replies
12
Views
905
Replies
5
Views
1K
  • General Math
Replies
23
Views
1K
Replies
1
Views
779
Replies
3
Views
1K
  • Precalculus Mathematics Homework Help
Replies
3
Views
478
  • General Math
Replies
4
Views
905
Replies
4
Views
253
  • General Math
Replies
6
Views
1K
  • Precalculus Mathematics Homework Help
Replies
10
Views
766
Back
Top