Is $f(x) =|x| - 3$ even, odd, or neither?

Question

$$f(x)=|x|-3$$

Is the function above odd, even or neither?

I know that a function is even if $f(x) = f(-x)$: $$f(-x) = |(-x)| - 3$$ $$f(-x) = x-3$$

Does this mean that the function is even? When I try to graph the function on my graphing calculator I get a syntax error.

Answer 1

Your logic is correct, and you have proved that the function is even.

To draw a graph of the function, all you need to do is to draw the graph $y=|x|$ and shift it down by $3$ for the $-3$ factor in the equation.

Here is the graph:

Answer 2

We know that $|x|=|-x|$, for all $x\in\Bbb R$. Therefore, $|x|-3=|-x|-3$, which is equivalent to $f(x)=f(-x)$. So yes, you are right. It is even.