Answered

Which statement best describes how to determine whether [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is an even function?

A. Determine whether [tex]\( -\left( x^3 + 5x + 1 \right) \)[/tex] is equivalent to [tex]\( x^3 + 5x + 1 \)[/tex].
B. Determine whether [tex]\( (-x)^3 + 5(-x) + 1 \)[/tex] is equivalent to [tex]\( x^3 + 5x + 1 \)[/tex].
C. Determine whether [tex]\( -x^3 + 5x + 1 \)[/tex] is equivalent to [tex]\( -\left( x^3 + 5x + 1 \right) \)[/tex].
D. Determine whether [tex]\( (-x)^3 + 5(-x) + 1 \)[/tex] is equivalent to [tex]\( -\left( x^3 + 5x + 1 \right) \)[/tex].



Answer :

GinnyAnswer
To determine whether the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is even, we need to check if it satisfies the definition of an even function. A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex].

Let's go through the steps to check this:

1. Substitute [tex]\( -x \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-x) = (-x)^3 + 5(-x) + 1 \][/tex]

2. Simplify [tex]\( f(-x) \)[/tex]:
[tex]\[ (-x)^3 + 5(-x) + 1 = -x^3 - 5x + 1 \][/tex]

3. Compare [tex]\( f(-x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[ f(-x) = -x^3 - 5x + 1 \][/tex]
[tex]\[ f(x) = x^3 + 5x + 1 \][/tex]

4. Determine whether [tex]\( f(-x) \)[/tex] is equivalent to [tex]\( f(x) \)[/tex]:
[tex]\[ -x^3 - 5x + 1 \neq x^3 + 5x + 1 \][/tex]

Since [tex]\( f(-x) \)[/tex] is not equal to [tex]\( f(x) \)[/tex], the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is not an even function.

Thus, the best statement that describes how to determine whether [tex]\( f(x) = x^3 + 5 x + 1 \)[/tex] is an even function is:
- Determine whether [tex]\( (-x)^3 + 5(-x) + 1 \)[/tex] is equivalent to [tex]\( x^3 + 5 x + 1 \)[/tex].

So, the correct option is:
- Determine whether [tex]\((-x)^3 + 5(-x) + 1\)[/tex] is equivalent to [tex]\(x^3 + 5 x + 1\)[/tex].

Other Questions