Question

A polynomial function has the form f(x)=axn+bx, where a and b are nonzero constants, and n is a nonnegative integer. Which of the following conditions guarantees that f(x) is an even function?

a) a=0
b) n is even
c) n is odd
d) b=0

Answer 1

An even function is a function where f(x) = f(-x) for all x. In the given polynomial function f(x) = ax^n + bx, the condition that guarantees f(x) is an even function is n is even.

Step-by-step explanation:

An even function is a function where f(x) = f(-x) for all x. In the given polynomial function f(x) = ax^n + bx, there are two conditions that guarantee that f(x) is an even function:

1. a = 0. If a = 0, then f(x) simplifies to bx, which is an odd function.
2. n is even. If n is even, then f(x) simplifies to ax^n + bx = ax^n + bx = -ax^n + bx = -f(x), satisfying the condition for an even function.

Therefore, the answer is option b) n is even.