Let f(x) = 1 – 3x^2. Which of the following is true? Please give us a brief explanation.
A. f is an odd function.
B. f is an even function.
C. f is neither odd nor even.
D. f is both odd and even.
Your choices are missing.
To determine whether the function f(x) = 1 - 3x^2 is odd, even, or neither, we need to check whether it satisfies the properties of odd and even functions:
1. Odd function: f(x) is odd if f(-x) = -f(x) for all x in the domain of f.
2. Even function: f(x) is even if f(-x) = f(x) for all x in the domain of f.
Now let's evaluate f(-x):
f(-x) = 1 - 3(-x)^2
= 1 - 3x^2
Comparing f(-x) and -f(x):
-f(x) = -1 + 3x^2
We can see that f(-x) ≠ -f(x), which means f(x) = 1 - 3x^2 does not satisfy the condition for an odd function.
Next, let's check whether f(-x) = f(x):
f(-x) = f(x)
1 - 3(-x)^2 = 1 - 3x^2
1 - 3x^2 = 1 - 3x^2
Here we can see that f(-x) = f(x), which means f(x) = 1 - 3x^2 satisfies the condition for an even function.
Therefore, the correct answer is B. f is an even function.