Are the graphs of f(|x|) and f(|−x|) same or not?

Question

Graph of $f(x)$ :

Original question says that the graph of $f(x)$ is given above. According to this , which of the following functions have continuous graphs in real numbers?

I-) $|f(x)|$

II-) $|-f(x)|$

III-) $f(|x|)$

IV-) $f(|-x|)$

Answer: I,II,IV

What I know :

• $|x|=|-x|$

• $f(|x|)$ reflects the graph to the right of the y-axis in the y-axis. Ignore the left hand side part of the graph

What am I asking ? : Because of $|x|=|-x|$ , i thought that the graph of $f(|x|)$ and $f(|-x|)$ must be same ,and because of the fact that " $f(|x|)$ reflects the graph to the right of the y-axis in the y-axis. Ignore the left hand side part of the graph " , i thought that both $f(|x|)$ and $f(|-x|)$ have discontinuous graphs.

However , the given answer is $I,II,IV$.According to this answer , the graph of $f(|x|)$ and $f(|-x|)$ are different , but i do not think so. (I agree with I and II but not IV)

Can you explain me why the graph of $f(|x|)$ and $f(|-x|)$ are different ?

NOTE: When i asked it to my teacher , he said that $f(|-x|)$ reflects the graph to the LEFT of the y-axis in the y-axis ,so IV is continuous but it does not make sense to me . What do you think ?

Answer 1

$f(|x|)$ and $f(|-x|)$ have the same graphs because they're the same function. The answer key is wrong; $f(|x|)$ is not continuous (particularly there's a discontinuity at $2$). Perhaps they have a typo and intended option IV to be $f(-|x|)$; in this case it is continuous and would be a valid choice.