Determine if function is even, odd, or neither...
f(x) = secx tanx
I understand how to find if its even or odd, but the sec and tan, I don't understand.
Can someone help?
Isn't sec even (reciprocal of cosine). And Tan is odd, so the product is...
You can also graph it on your graphical calc.
To determine if the function f(x) = sec(x) tan(x) is even, odd, or neither, we need to understand the properties of even and odd functions.
An even function is symmetric about the y-axis, meaning that if you reflect the graph of the function across the y-axis, it remains unchanged. Mathematically, for an even function f(x), it must satisfy the condition f(x) = f(-x) for all x.
An odd function is symmetric about the origin, meaning that if you reflect the graph of the function across the origin, it remains unchanged. Mathematically, for an odd function f(x), it must satisfy the condition f(x) = -f(-x) for all x.
Now let's examine the properties of sec(x) and tan(x):
1. Secant function (sec(x)): The secant function is defined as sec(x) = 1/cos(x). It is not an even or odd function because sec(-x) = 1/cos(-x) ≠ sec(x) and sec(-x) ≠ -sec(x). So, sec(x) is neither even nor odd.
2. Tangent function (tan(x)): The tangent function is defined as tan(x) = sin(x)/cos(x). Similar to the secant function, tan(x) is neither even nor odd because tan(-x) = sin(-x)/cos(-x) ≠ tan(x) and tan(-x) ≠ -tan(x).
Now, let's consider their product, f(x) = sec(x) tan(x). To determine if f(x) is even or odd, we need to check if f(x) satisfies the conditions for even or odd functions.
- Even function: We need to check if f(x) = f(-x) for all x. Let's substitute -x into f(x) and check if it is equal to f(x):
f(-x) = sec(-x) tan(-x)
= sec(x) (-tan(x)) <-- multiply by -1
= -sec(x) tan(x)
Since f(-x) = -sec(x) tan(x) ≠ f(x), we can conclude that f(x) is not an even function.
- Odd function: We need to check if f(x) = -f(-x) for all x. Let's substitute -x into f(x) and check if it is equal to -f(x):
-f(-x) = -sec(-x) tan(-x)
= -sec(x) (-tan(x)) <-- multiply by -1
= sec(x) tan(x)
Since -f(-x) = sec(x) tan(x) = f(x), we can conclude that f(x) is an odd function.
Therefore, the function f(x) = sec(x) tan(x) is an odd function.