Let f and g be two odd functions. Prove that:
a) f + g is an odd function
b) g of f is an odd function
I am not even sure where to start, any help that can be provided would be appreciated!
A function f is odd iff
f(-x) = - f(x)
Put h(x) = f(x) + g(x) and calculate
h(-x):
h(-x) = f(-x) + g(-x) =
-f(x) - g(x) =
-[f(x) + g(x)] =
-h(x)
So, we see that h is odd because h(-x) = -h(x)
Now put h(x) = g[f(x)]
h(-x) = g[f(-x)] =
g[-f(x)] =
-g[f(x)] =
-h(x)
And we see that h is odd.
One more exercise you could do:
If f(x) is an arbitrary function show that it can be decomposed uniquely as:
f(x) = f_even(x) + f_odd(x)
where f_even and f_odd are even and odd functions, respectively. Give the expressions for these functions in terms of the function f.
got it.
Thank you so much!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
To prove that the sum of two odd functions is odd, we need to show that for any value x, (f + g)(-x) = -(f + g)(x). Here's how you can approach the proof:
a) Proving (f + g) is an odd function:
1. Start with the definition of an odd function: f(x) = -f(-x) and g(x) = -g(-x) for all values of x.
2. Consider the sum of the two functions: (f + g)(x) = f(x) + g(x).
3. Substitute -x for x in the sum: (f + g)(-x) = f(-x) + g(-x).
4. Using the property of odd functions, we can rewrite f(-x) and g(-x) as -f(x) and -g(x): (f + g)(-x) = -f(x) - g(x).
5. Rearrange the expression: (f + g)(-x) = -(f(x) + g(x)).
6. Based on step 5, we have shown that (f + g)(-x) = -(f + g)(x), which verifies that f + g is an odd function.
b) Proving g(f) is an odd function:
1. Start with the definition of an odd function: f(x) = -f(-x) and g(x) = -g(-x) for all values of x.
2. Consider the composite function: (g ∘ f)(x) = g(f(x)).
3. Substitute -x for x in the composite function: (g ∘ f)(-x) = g(f(-x)).
4. Using the property of odd functions, we can rewrite f(-x) as -f(x): (g ∘ f)(-x) = g(-f(x)).
5. Also, we can rewrite g(-x) as -g(x) based on the property of odd functions: (g ∘ f)(-x) = -g(f(x)).
6. Based on step 5, we have shown that (g ∘ f)(-x) = -(g ∘ f)(x), which verifies that g(f) is an odd function.
By following these steps, you can prove that both f + g and g(f) are odd functions.