If [tex]$g(x)$[/tex] is an odd function, which function must be an even function?
[tex]$f(x)=g(x)+2$[/tex]
[tex]$f(x)=g(x)+g(x)$[/tex]
[tex]$f(x)=g(x)^2$[/tex]
[tex]$f(x)=-g(x)$[/tex]
Answer (2)
An odd function g ( x ) satisfies g ( − x ) = − g ( x ) .
Check each option to see if f ( − x ) = f ( x ) (even function).
f ( x ) = g ( x ) 2 satisfies f ( − x ) = g ( − x ) 2 = ( − g ( x ) ) 2 = g ( x ) 2 = f ( x ) .
Therefore, the even function is f ( x ) = g ( x ) 2 .
Explanation
Understanding Odd and Even Functions We are given that g ( x ) is an odd function. This means that g ( − x ) = − g ( x ) for all x in the domain of g . We need to determine which of the given functions f ( x ) must be an even function. A function f ( x ) is even if f ( − x ) = f ( x ) for all x in the domain of f . Let's examine each option.
Analyzing Option 1 Option 1: f ( x ) = g ( x ) + 2 . Then f ( − x ) = g ( − x ) + 2 = − g ( x ) + 2 . This is not necessarily equal to f ( x ) , so f ( x ) is not necessarily even. For example, if g ( x ) = x , then f ( x ) = x + 2 and f ( − x ) = − x + 2 , which is not equal to f ( x ) .
Analyzing Option 2 Option 2: f ( x ) = g ( x ) + g ( x ) = 2 g ( x ) . Then f ( − x ) = 2 g ( − x ) = 2 ( − g ( x )) = − 2 g ( x ) = − f ( x ) . This means f ( x ) is an odd function, not an even function.
Analyzing Option 3 Option 3: f ( x ) = g ( x ) 2 . Then f ( − x ) = g ( − x ) 2 = ( − g ( x ) ) 2 = g ( x ) 2 = f ( x ) . Therefore, f ( x ) is an even function.
Analyzing Option 4 Option 4: f ( x ) = − g ( x ) . Then f ( − x ) = − g ( − x ) = − ( − g ( x )) = g ( x ) = − f ( x ) . This means f ( x ) is an odd function, not an even function.
Conclusion Therefore, the only function that must be an even function is f ( x ) = g ( x ) 2 .
Examples
Understanding even and odd functions helps in analyzing symmetrical properties in various fields, such as physics and engineering. For instance, when studying the behavior of waves or signals, recognizing even or odd symmetry can simplify the analysis and prediction of their behavior. In signal processing, even functions are often associated with cosine waves, while odd functions are associated with sine waves. Recognizing these symmetries can help in designing filters and analyzing frequency content.
The function that must be an even function given that g ( x ) is odd is f ( x ) = g ( x ) 2 since it satisfies the condition f ( − x ) = f ( x ) . Other options either result in odd functions or expressions that do not meet the even condition. Thus, the correct choice is f ( x ) = g ( x ) 2 .
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