Determine if Odd, Even, or Neither f(x)=|x|

Problem

ƒ(x)=|x|

Solution

1. Recall the definitions of even and odd functions. A function is even if ƒ*(−x)=ƒ(x) and odd if ƒ*(−x)=−ƒ(x) for all x in the domain.

2. Substitute −x into the function to evaluate ƒ*(−x)

ƒ*(−x)=|−x|

3. Apply the property of absolute value, which states that the distance of a number from zero is the same as the distance of its opposite from zero, meaning |−a|=|a|

|−x|=|x|

4. Compare the result to the original function. Since |x|=ƒ(x) we have ƒ*(−x)=ƒ(x)

ƒ*(−x)=ƒ(x)

5. Conclude that because ƒ*(−x)=ƒ(x) the function is even.

Final Answer

ƒ(x)=|x|* is Even

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