How do you find whether a function is even or odd?
Identify the odd function.
A)f(x) = 5
B)f(x) = 8x
C)f(x) = |x3|
D)f(x) = x2 + 7
To determine whether a function is even or odd, you need to consider the behavior of the function when its argument changes signs.
1. Start with the definition:
- An even function is one where f(x) = f(-x) for all x in the domain.
- An odd function is one where f(x) = -f(-x) for all x in the domain.
2. For an even function:
- Substitute -x for x in the function and simplify.
- If the resulting expression is the same as the original function, then the function is even.
3. For an odd function:
- Substitute -x for x in the function and simplify.
- If the resulting expression is the negative of the original function, then the function is odd.
Now let's analyze the given functions to identify the odd function:
A) f(x) = 5
- Substitute -x for x: f(-x) = 5
- Since f(-x) = f(x), this function is even.
B) f(x) = 8x
- Substitute -x for x: f(-x) = 8(-x) = -8x
- Since f(-x) = -f(x), this function is odd.
C) f(x) = |x^3|
- Substitute -x for x: f(-x) = |-x^3| = |-(x^3)| = -|x^3|
- Since f(-x) = -f(x), this function is odd.
D) f(x) = x^2 + 7
- Substitute -x for x: f(-x) = (-x)^2 + 7 = x^2 + 7
- Since f(-x) = f(x), this function is even.
From this analysis, we can conclude that the odd function is B) f(x) = 8x.