Question about bijections Determine whether or not the following functions from real numbers to real numbers are bijections. If they are
bijections, then find the inverse. If they are not bijections, then explain why not.
I am trying to determine whether some of these function listed below are bijections or not. I understand that the functions are from real to real numbers. Any input on how to go about figuring this out? Thanks!
the two functions are 
$f\left(x\right) = 5x$
and
$f\left(x\right) = \left|4x-12\right|$
thanks in advance!
 A: You need to ask yourself in each case: Is the function an injection (one-one)? is it a surjection (for every real number $y$, is there an $x$ such that $f(x) = y$?


*

*The answer to both questions for $f(x) = 5x$ is evidently positive -- why?

*Note that in the second case every value of $f$ is positive: how does that impact on the question of surjectivity?
A: f^{-1}\left(x\right)=\frac{x}{5}
For both $x=2$ and $x=4$ there's equal values of function $f\left(x\right)=4$ so there's no inverse function - because then you would have two different values for $f^{-1}\left(4\right)$.
A: $f(x) = 5x$
If you can find $g$ so that $g(f(x))=x$ then $f$ is a bijection. To do this, you express $x$ in terms of $f(x)$.
$f(x) = 5x$
Divide both sides by $5$: $x=\frac{f(x)}{5}$
So $g(x)=\frac{x}{5}$ works which means $f$ is a bijection.

Then there are more complicated cases where you need to prove it is a bijection (or not) but don't have to explicitly give a inverse function. In those base, you use the fact that $f$ is bijective iff it is injective and surjective.
$f(x) = |4x-12|$
If you think about $x\mapsto |4x-12|$, you see that it obviously isn't a bijection since the image is positive for any given number. Therefore, you can just pick any negative number such as $-1$ and say that it doesn't have any antecedent by $f$ so $f$ is not surjective and therefore not bijective.
A: As said in the comments, you have to check injectivity and surjectivity. 
For injectivity you must show that for every $x,y\in\mathbb{R}$, if $x\neq y$ then also $f(x)\neq f(y)$, i.e. it maps different values to different values.
For surjectivity you must show that for every $y\in\mathbb{R}$ you can find an $x\in\mathbb{R}$ such that $f(x)=y$, i.e. you can hit everything with $f$.
Let's take $f(x)=5x$. Suppose $x\neq y$. Is it then true that also $f(x)=5x\neq 5y=f(y)$? This answers the injectivity. Furthermore if $y\in\mathbb{R}$ is arbitrary, can you find an $x\in\mathbb{R}$ such that $5x=f(x)=y$. Hint: solve the equation $5x=y$ for $x$. This answers the surjectivity. 
