Which statement best describes how to determine whether [tex]f(x)=x^2=x+8[/tex] is an even function?

A. Determine whether [tex]-x^2-(-x)+8[/tex] is equivalent to [tex]x^2-x+8[/tex].
B. Determine whether [tex](-x)^2-(-x)+8[/tex] is equivalent to [tex]x^2-x+8[/tex].
C. Determine whether [tex]-x^2-(-x)+8[/tex] is equivalent to [tex]-(x^2-x+8)[/tex].
D. Determine whether [tex](-x)^2=(-x)+8[/tex] is equivalent to [tex]-(x^2-x+8)[/tex].

To check if f ( x ) is even, verify if f ( − x ) = f ( x ) .
Substitute − x into f ( x ) to find f ( − x ) = ( − x ) 2 − ( − x ) + 8 = x 2 + x + 8 .
Determine if x 2 + x + 8 is equivalent to x 2 − x + 8 .
The statement that best describes this process is: Determine whether ( − x ) 2 − ( − x ) + 8 is equivalent to x 2 − x + 8 .

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 this case, f ( x ) = x 2 − x + 8 . We need to find f ( − x ) and see if it is equal to f ( x ) .

Finding f(-x) To find f ( − x ) , we substitute − x for x in the expression for f ( x ) : f ( − x ) = ( − x ) 2 − ( − x ) + 8 Now, we simplify this expression.

Simplifying f(-x) Simplifying f ( − x ) , we get: f ( − x ) = ( − x ) 2 − ( − x ) + 8 = x 2 + x + 8 Now we need to check if f ( − x ) = f ( x ) , which means we need to check if x 2 + x + 8 = x 2 − x + 8 .

Comparing f(-x) and f(x) The statement that best describes how to determine whether f ( x ) = x 2 − x + 8 is an even function is the one that compares ( − x ) 2 − ( − x ) + 8 with x 2 − x + 8 . This is because we are substituting − x into the function and checking if the result is the same as the original function.

Final Answer The correct statement is: Determine whether ( − x ) 2 − ( − x ) + 8 is equivalent to x 2 − x + 8 .
This is because it directly checks if f ( − x ) = f ( x ) .


Examples
Even functions are symmetric about the y-axis. For example, the cost of electricity might be modeled as an even function if the cost is the same whether you use the electricity during the day or at night. Checking if a function is even helps us understand its symmetry and predict its behavior. For instance, if a bridge's structure is designed symmetrically, the load distribution can be modeled using even functions to ensure stability on both sides.

Answered by GinnyAnswer | 2025-07-03

To determine if f ( x ) = x 2 − x + 8 is an even function, we check if f ( − x ) = f ( x ) . Upon finding f ( − x ) = x 2 + x + 8 , we see it does not equal f ( x ) , confirming that the function is not even. Therefore, the best statement is option B, which involves checking ( − x ) 2 − ( − x ) + 8 .
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Answered by Anonymous | 2025-07-04