How do you determine if # f(x)=x/(x+1)# is an even or odd function?
1 Answer
Explanation:
To determine if the given function is even or odd, you must prove the following cases:
#1# . If the function is even, then#f(-x)=f(x)# .
#2# . If the function is odd, then#f(-x)=-f(x)# .
Note that if the function is neither even nor odd, then it is neither.
Case 1 - Even Test
Determine the left and right sides of the given function using
#"LS"=f(-x)color(white)(XXXXXXXXX)"RS"=f(x)#
#"LS"=((-x))/((-x)+1)color(white)(XXXXXXX)"RS"=color(blue)(|bar(ul(color(white)(a/a)color(black)(x/(x+1))color(white)(a/a)|)))#
#"LS"=color(blue)(|bar(ul(color(white)(a/a)color(black)((-x)/(-x+1))color(white)(a/a)|)))# Since
#"LS"# #!=# #"RS"# ,#f(x)# is not even.
Case 2 - Odd Test
Determine the left and right sides of the given function using
#"LS"=f(-x)color(white)(XXXXXXXXX)"RS"=-f(x)#
#"LS"=((-x))/((-x)+1)color(white)(XXXXXXX)"RS"=-(x/(x+1))#
#"LS"=color(blue)(|bar(ul(color(white)(a/a)color(black)((-x)/(-x+1))color(white)(a/a)|)))color(white)(XXXXxx)"RS"=color(blue)(|bar(ul(color(white)(a/a)color(black)((-x)/(x+1))color(white)(a/a)|)))# Since
#"LS"# #!=# #"RS"# ,#f(x)# is not odd.
The Conclusion
From the even and odd tests, since the given function is neither even nor odd, it is neither.
#:.# ,#f(x)# is neither.