Determining whether a function is injective or surjective in a certain range

functions

I ran into this question while preparing for an exam:

Let f : [1, 3] → [0, 2] be the function defined by
f(x) = ln x
for all x ∈ [1, 3]. Determine whether or not this function is injective
and whether or not it is surjective

I understand what injective (one-to-one) and surjective (onto) are but I don't know where to start to show surjectivity and the specified range is confusing me also as I know functions alone can't be injective, bijective, or surjective as it is dependant on the range

This is my attempt so far to show injectivity but I am not sure it is correct:

y=logx is injective.

x1=x2⟹logx1=lnx2

logx1=logx2⟹x1=x2

This is the graph I obtained from desmos:
enter image description here

Any help would be appreciated thank you!

Best Answer

We assume that the properties of the function $\ln:\mathbb R^{+*}\to \mathbb R$ are known.

Then your proof for injectivity is correct.

For surjectivity, thanks to the remarks that have been made to you, rather than a simple graphical reading or the use of the calculator, as you are interested in proofs, I suggest this one:

$2\in [0,2]$. Let us prove that there is no element $x\in [1,3]$ such that $\ln(x)=2$. As the funtion is an increasing function, it suffices to show that $2>\ln(3)$.

$e>2$. So, $e^2>4>3.$ So, $\ln(e^2)>\ln(3)$, i.e. $2>\ln(3)$. Q.E.D.

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