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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].

Sagot :

GinnyAnsweridn
To determine whether the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] is even, you need to compare [tex]\( f(x) \)[/tex] with [tex]\( f(-x) \)[/tex].

A function [tex]\( f(x) \)[/tex] is even if:

[tex]\[ f(x) = f(-x) \][/tex]

Starting with [tex]\( f(-x) \)[/tex], let's substitute [tex]\(-x\)[/tex] into the function [tex]\( f(x) = x^3 + 5x + 1 \)[/tex] and simplify:

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

Simplifying [tex]\( f(-x) \)[/tex]:

[tex]\[ (-x)^3 = -x^3 \][/tex]
[tex]\[ 5(-x) = -5x \][/tex]

So:

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

Now we 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]

Clearly, [tex]\( f(-x) \)[/tex] is not equivalent to [tex]\( f(x) \)[/tex] because:

[tex]\[ x^3 + 5x + 1 \neq -x^3 - 5x + 1 \][/tex]

This means that [tex]\( f(x) \)[/tex] is not an even function.

Therefore, the statement that best describes how to determine whether [tex]\( f(x) \)[/tex] is even is:

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

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