Determining if a function is odd or even

The Attempt at a Solution

$$f(-x)=\begin{cases} -x+5,\quad -2<x<0 \\ x+5,\quad 0≤x<2 \end{cases}\\ =f(x)$$

The issue is that I can't get to the second step. I know the function is even.

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pasmith
Homework Helper

The Attempt at a Solution

$$f(-x)=\begin{cases} -x+5,\quad -2<x<0 \\ x+5,\quad 0≤x<2 \end{cases}\\ =f(x)$$

The issue is that I can't get to the second step. I know the function is even.
You should have
$$f(-x)=\begin{cases} -x+5,\quad -2<-x<0 \\ x+5,\quad 0≤-x<2 \end{cases}$$

(You need to substitute -x for x in the inequalities as well.)

HallsofIvy
Homework Helper

The Attempt at a Solution

$$f(-x)=\begin{cases} -x+5,\quad -2<x<0 \\ x+5,\quad 0≤x<2 \end{cases}\\ =f(x)$$

The issue is that I can't get to the second step. I know the function is even.
I have no idea what you mean by "the second step". (What was the first step?) Just use the definition of "even function".

If -2< x< 0 then 0< x< 2 so f(-x)= (-x)+ 5= -x+ 5= f(x).
If 0< x< 2 then -2< x< 0 so f(-x)= -(-x)+ 5= x+ 5= f(x).