Question

complete parts a through d to follow the step - by - step procedure to determine if the following function is even, odd, or neither. f(x)=5x^4 - x^2 + 3 d. f( - x)=f(x) c) determine f( - x) and simplify. f( - x)=□ (simplify your answer. do not factor) d) is the given function even, odd, or neither? a. the function is even b. the function is odd c. the function is neither even nor odd.

Explanation:

Step1: Substitute -x into the function

Given \(f(x)=5x^{4}-x^{2}+3\), substitute \(x = -x\).
\[f(-x)=5(-x)^{4}-(-x)^{2}+3\]

Step2: Simplify the expression

Since \((-x)^{4}=x^{4}\) and \((-x)^{2}=x^{2}\), we have:
\[f(-x)=5x^{4}-x^{2}+3\]

Step3: Compare f(-x) with f(x)

We know \(f(x)=5x^{4}-x^{2}+3\) and \(f(-x)=5x^{4}-x^{2}+3\), so \(f(-x)=f(x)\). A function is even if \(f(-x)=f(x)\), odd if \(f(-x)=-f(x)\). Since \(f(-x)=f(x)\), the function is even.

Answer:

c) \(f(-x)=5x^{4}-x^{2}+3\)
d) A. The function is even