Given a function $f(x)$ that verifies the following conditions

Question

I have a function that verifies the following conditions:

$f(x)$ is even

$f(5)=6$

$f(x)$ belongs to $[0,5)$ for $x \in [-2,-1]$

It increases in $(- \infty ,-6)$

$\lim_{x\to 6+ } = + \infty$

The activity says:

1) Are the following statements true?:

I) $0 \le f(1,5) \le 6 $

II) The limit $f(x)$ when $x$ approaches 5 from the right is 6

2) What can you say about the evenness or oddness of $f(3x), f(x+3), 3f(x) and f(x)+3$ ?

I need some hints to do it.

Answer 1

Statement II is false, as $f(5)=6$ but this gives us no information about the limits (it would if we knew $f$ is continuous, but we are making no such assumption) I don't really understand what I is really asking. For the rest:

$f(3x) = f(-3x) $ because $f$ is even. So, if $g(x)=f(3x)$, $g$ is an even function $f(x+3) =f(-x-3)$ which does not necessarelly equal $f(-x+3)$, so you will easily find a counterexample on that one. The remaining are very easy reasoning in a similar way