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 into the equation and got the answers:
(1)= 2k
(-1)= k
A function is even if f(a) = f(-a)
let's test it
f(a) = a^4 + ka^2 + 1
f(-a) = (-a)^4 + k(-a)^2 + 1 = a^4 + ka^2 + 1
so, what do you think?
That they are correct, this is an even function ?
To determine if a function is even, we need to check if f(x) is equal to f(-x) for all values of x.
Let's substitute -x into the function and check if f(-x) = f(x):
f(-x) = (-x)^4 + k(-x)^2 + 1
= x^4 + kx^2 + 1
Since f(-x) is equal to f(x), the function f(x) is indeed an even function.
This means that the student's claim is true for all values of k, so the correct statement is:
The student's claim is true for all values of k.
To determine whether the function f(x) = x^4 + kx^2 + 1 is an even function, we need to understand the concept of even functions.
An even function is a function for which f(x) = f(-x) for all x in the domain. In simpler terms, if you replace x with -x and the function remains unchanged, then it is an even function.
To check if f(x) = x^4 + kx^2 + 1 is even, substitute -x for x in the function:
f(-x) = (-x)^4 + k(-x)^2 + 1
= x^4 + kx^2 + 1
As you can see, f(-x) is equal to f(x), indicating that the function is unchanged when replaced by its negation. Hence, the function f(x) = x^4 + kx^2 + 1 is indeed an even function for all values of k.
Therefore, the statement "The student's claim is true for all values of k" is the correct answer.