A student claims that the function f(x)=x4+kx2+1 is an even function. Which statement best describes the student's claim?
The answer is the first one for anyone who sees this. :) (well it was for me)
A student claims that the function f(x)=x^4 + kx^2 + 1 is an even function. Which statement best describes the student's claim?
The student's claim is true for all values of k.
The student's claim is only true for even values of k.
The student's claim is not true for any value of k.
The student's claim is only true for odd values of k.
I think the answer is the third one or fourth one but I am not sure
Because I substituted "x" with 1 and -1 and got
(1)=2k
(-1)= k
all powers of x are even, so the function is even for all k.
f(-x) = f(x) so f is even.
The student's claim that the function f(x) = x^4 + kx^2 + 1 is an even function can be evaluated by understanding the properties of even functions and analyzing the given function.
In mathematics, a function is considered even if it satisfies the property that f(x) = f(-x) for all values of x within its domain. In other words, an even function is symmetrical with respect to the y-axis.
To determine if the given function is even, you need to substitute (-x) into the function and see if the equation holds true. By doing this, you can determine if f(x) = f(-x) for all values of x.
Let's substitute (-x) into the function:
f(-x) = (-x)^4 + k(-x)^2 + 1
Simplifying this expression gives:
f(-x) = x^4 + kx^2 + 1
Comparing this with the original function f(x), which is x^4 + kx^2 + 1, we can see that f(x) = f(-x) for all values of x.
Therefore, the student's claim is correct, and the function f(x) = x^4 + kx^2 + 1 is an even function.