Show that f(-x) = -f(x)
Neat trick, but f(x) has to be an odd function for this to happen. Exactly what is the functions.
That is only for odd functions.
for example sin(-x) = -sin (x)
however
cos(-x) = + cos (x) so it is an even function
To prove that f(-x) = -f(x), we need to show that the function f is an odd function.
An odd function is a function that satisfies the property f(-x) = -f(x) for all values of x in its domain.
Here's how you can show that f(-x) = -f(x):
1. Start by assuming that f is an odd function.
2. Let x be any arbitrary value in the domain of the function f.
3. Evaluate f(-x) by substituting -x into the function f.
4. Evaluate -f(x) by substituting x into the function f and taking the negative of the result.
5. Simplify the expressions for f(-x) and -f(x) to see if they are equal. If they are equal, then f(-x) = -f(x) holds true for that particular x.
6. Repeat steps 2 to 5 for all values of x in the domain of the function to demonstrate that f(-x) = -f(x) holds true for all values of x in its domain.
If f(-x) = -f(x) holds true for all x in the domain of the function f, then we can conclude that f is an odd function.