Determine whether the function f(x)=sin^4xtanx is even or odd
To determine whether the function f(x) = sin^4(x)tan(x) is even or odd, we need to check if it satisfies the following properties:
1. Even function property: A function f(x) is even if f(x) = f(-x) for all x in its domain.
2. Odd function property: A function f(x) is odd if f(x) = -f(-x) for all x in its domain.
Let's check both properties for f(x) = sin^4(x)tan(x):
1. Even function property:
f(-x) = sin^4(-x)tan(-x)
Since sin(-x) = -sin(x) and tan(-x) = -tan(x), we have:
f(-x) = (-sin(x))^4 * (-tan(x))
= sin^4(x)tan(x)
Therefore, f(-x) = f(x) for all x in its domain.
2. Odd function property:
-f(-x) = -[sin^4(-x)tan(-x)]
= -[(-sin(x))^4 * (-tan(x))]
= sin^4(x)tan(x)
Therefore, -f(-x) = f(x) for all x in its domain.
Since f(x) = f(-x) and -f(-x) = f(x) for all x in the domain of f(x), the function f(x) = sin^4(x)tan(x) satisfies both the even and odd function properties. While it is possible for a function to be both even and odd (such as f(x) = 0), most functions are neither even nor odd.
Therefore, f(x) = sin^4(x)tan(x) is neither even nor odd.
To determine whether the function f(x) = sin^4(x)tan(x) is even or odd, we can use the following definitions:
- A function f(x) is even if f(x) = f(-x) for all x in the domain.
- A function f(x) is odd if f(x) = -f(-x) for all x in the domain.
Let's first check if f(x) = sin^4(x)tan(x) satisfies the condition for being an even function:
f(x) = sin^4(x)tan(x)
f(-x) = sin^4(-x)tan(-x)
Since sin(-x) = -sin(x) and tan(-x) = -tan(x), we can substitute these values:
f(-x) = (-sin(x))^4(-tan(x))
= sin^4(x)(-tan(x))
= -sin^4(x)tan(x)
= -f(x)
Since f(-x) = -f(x) for all x in the domain, the function f(x) = sin^4(x)tan(x) is an odd function.