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

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

To check if a function is even, you determine whether f ( − x ) is equivalent to f ( x ) . For the given function f ( x ) = x 3 + 5 x + 1 , this involves the following steps:

Substitute − x into the function: f ( − x ) = ( − x ) 3 + 5 ( − x ) + 1 .
Simplify the expression: f ( − x ) = − x 3 − 5 x + 1 .
Check if f ( − x ) is equivalent to f ( x ) : − x 3 − 5 x + 1 = x 3 + 5 x + 1 .
The statement that best describes this process is: Determine whether ( − x ) 3 + 5 ( − x ) + 1 is equivalent to x 3 + 5 x + 1 .

The final answer is: Determine whether ( − x ) 3 + 5 ( − x ) + 1 is equivalent to x 3 + 5 x + 1 .
Explanation

Understanding Even Functions To determine if a function f ( x ) is even, we need to check if f ( − x ) = f ( x ) for all x in the domain of f . In other words, replacing x with − x should result in the same function. We are given the function f ( x ) = x 3 + 5 x + 1 . We need to find the statement that correctly describes how to check if this function is even.

Analyzing the Options Let's analyze each option:

Determine whether − ( x 3 + 5 x + 1 ) is equivalent to x 3 + 5 x + 1 This is checking if − f ( x ) = f ( x ) , which is related to odd functions, not even functions.

Determine whether ( − x ) 3 + 5 ( − x ) + 1 is equivalent to x 3 + 5 x + 1 This is checking if f ( − x ) = f ( x ) , which is the correct condition for even functions.

Determine whether − x 3 + 5 x + 1 is equivalent to − ( x 3 + 5 x + 1 ) .
This is comparing − x 3 + 5 x + 1 with − x 3 − 5 x − 1 , which is not relevant to checking for even functions.

Determine whether ( − x ) 3 + 5 ( − x ) + 1 is equivalent to − ( x 3 + 5 x + 1 ) This is checking if f ( − x ) = − f ( x ) , which is the condition for odd functions.

Finding the Correct Statement The correct statement is the one that checks if f ( − x ) = f ( x ) . We compute f ( − x ) :


f ( − x ) = ( − x ) 3 + 5 ( − x ) + 1 = − x 3 − 5 x + 1
Now we compare f ( − x ) with f ( x ) :
f ( x ) = x 3 + 5 x + 1
Since − x 3 − 5 x + 1 is not equal to x 3 + 5 x + 1 , the function is not even. However, the question asks for the correct statement to determine if the function is even, not whether the function is even.

Conclusion Therefore, the statement that best describes how to determine whether f ( x ) = x 3 + 5 x + 1 is an even function is:

Determine whether ( − x ) 3 + 5 ( − x ) + 1 is equivalent to x 3 + 5 x + 1
Examples
Even functions are symmetric about the y-axis. For example, the cost of electricity might be modeled as an even function if the price is the same regardless of whether you use the electricity during the day (positive x) or at night (negative x). Checking if a function is even helps determine if there's symmetry in the relationship being modeled, which can simplify analysis and predictions.

Answered by GinnyAnswer | 2025-07-03

To determine if f ( x ) = x 3 + 5 x + 1 is even, we check if f ( − x ) = f ( x ) . The correct method is option B: check if ( − x ) 3 + 5 ( − x ) + 1 equals x 3 + 5 x + 1 . Since the function does not satisfy this, it is not even.
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Answered by Anonymous | 2025-07-04