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]
Answer (2)
To determine if the function f ( x ) = x 2 − x + 8 is even, we calculate f ( − x ) and find that it equals x 2 + x + 8 . Since this is not equal to f ( x ) , the function is not even. The correct approach is described in option B: Check if ( − x ) 2 − ( − x ) + 8 equals x 2 − x + 8 .
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To check if f ( x ) is even, verify if f ( − x ) = f ( x ) .
Calculate f ( − x ) by substituting − x into f ( x ) , resulting in f ( − x ) = ( − x ) 2 − ( − x ) + 8 = x 2 + x + 8 .
Compare f ( − x ) = x 2 + x + 8 with f ( x ) = x 2 − x + 8 .
Determine if ( − x ) 2 − ( − x ) + 8 is equivalent to x 2 − x + 8 . The correct statement 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 other words, replacing x with − x in the function should result in the same function.
Calculating f(-x) Let's find f ( − x ) for the given function f ( x ) = x 2 − x + 8 . We substitute − x for x in the expression: f ( − x ) = ( − x ) 2 − ( − x ) + 8
Simplifying f(-x) Now, let's simplify the expression for f ( − x ) : f ( − x ) = ( − x ) 2 − ( − x ) + 8 = x 2 + x + 8
Comparing f(-x) and f(x) We need to check if f ( − x ) = x 2 + x + 8 is equivalent to f ( x ) = x 2 − x + 8 . Since x 2 + x + 8 is not the same as x 2 − x + 8 , the function is not even. The correct statement to determine if f ( x ) is even is to check if ( − x ) 2 − ( − x ) + 8 is equivalent to x 2 − x + 8 .
Final Answer Therefore, the statement that best describes how to determine whether f ( x ) = x 2 − x + 8 is an even function is: Determine whether ( − x ) 2 − ( − x ) + 8 is equivalent to x 2 − x + 8 .
Examples
Even functions are symmetric about the y-axis. In real life, you might see this symmetry in the design of a bridge or a building. If the left and right sides mirror each other across a central line, that's a visual representation of an even function. Understanding even functions helps engineers and designers create balanced and aesthetically pleasing structures.
