Confused Please Help! Thanks!
Let f(x) = 1 – 3x^2. Which of the following is true? Give 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
What are the choices?
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
Totally confused and would appreciate some insight. Thanks!
an even function is one where f(x) = f(-x), that is, using either a positive number as input to the function you get the same result as if you used the opposite of that number
and odd function is where f(x) = -f(x)
so what if you tried f(5) and f(-5)
f(5) = 1 - 3(5)^2 = 1 - 75 = -74
f(-5) = 1 - 3(-5)^2 = 1 - 75 = -74
will that be true for all ± x's ?
mmmhhh?
another way to look at it:
an even function results in a reflection in the y-axis
an odd function results in a reflection in the origin.
So I would say that the answer would be that f can be both odd and even then. Am I correct?
NO, no, no
it is EVEN.
Just make a quick sketch of the graph.
Did you not read my explanations ?
Yes I read them, but misinterpreted them I suppose.
So this is an even function because x is being represented rather than y and this would be an even function resulting in a reflection in the y-axis?
I will sketch it out.
If this was a -f(x) then it would be odd? I think I understand now..Am I correct?
To determine whether the function f(x) = 1 - 3x^2 is odd, even, or neither, we need to understand the definitions of odd and even functions.
1. Odd Function: A function f(x) is considered odd if f(-x) = -f(x) for every x in the domain of the function.
2. Even Function: A function f(x) is considered even if f(-x) = f(x) for every x in the domain of the function.
Now, let's apply these definitions to the given function f(x) = 1 - 3x^2:
1. Check for oddness:
To verify if f(x) is odd, we need to confirm whether f(-x) = -f(x) for every x in the domain.
Let's evaluate f(-x):
f(-x) = 1 - 3(-x)^2
= 1 - 3x^2
Since f(-x) = 1 - 3x^2 is not equal to -f(x) = -[1 - 3x^2] = -1 + 3x^2, we can conclude that f(x) is not an odd function.
2. Check for evenness:
To verify if f(x) is even, we need to confirm whether f(-x) = f(x) for every x in the domain.
Let's evaluate f(-x):
f(-x) = 1 - 3(-x)^2
= 1 - 3x^2
Since f(-x) = 1 - 3x^2 is equal to f(x) = 1 - 3x^2, we can conclude that f(x) is an even function.
Therefore, the correct answer is B. f is an even function.
Explanation: The function f(x) = 1 - 3x^2 is symmetric about the y-axis, meaning its graph is the same on both sides. This symmetry is the characteristic of an even function.