To determine if a function is even, odd or neither, we need to verify by the definition of an odd and even function, as follows:
Even function:
[tex]f(x)=f(-x)[/tex]
Odd function:
[tex]g(-x)=-g(x)[/tex]
In the number 9, we have the following function:
[tex]h(x)=|x|-1[/tex]
If we substitute the value from x to -x, we have the following:
[tex]h(-x)=|-x|-1[/tex]
but, by definition, we have:
[tex]|-x|=|-1\times x|=|-1|\times|x|=1\times|x|=|x|[/tex]
From this, we can rewrite the function h(-x) as follows:
[tex]h(-x)=|-x|-1=|x|-1=h(x)[/tex]
And from this, we can say that:
[tex]h(-x)=h(x)[/tex]
And from the solution developed above, we are able to conclude that the function described by h(x) in number 9 is an even function
