Injective and surjective proof. Let  $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=-2x^2+7x+15$. 

a) Is $f$ injection or surjection? 

b) Determine $f(\mathbb{R})$. 

c) How much results has the Equation $f (x) = b$ as a function of $b\in\mathbb{R}$?

d) Enter two largest possible domains $A\subset\mathbb{R}$ and matching image regions $B \subset\mathbb{R}$ so that $f : A \to B$ is bijective and determine the respective inverse function.

Can someone please explain me how to see if function is injection or surjection when we have x^2. What means to determine f(R) and how to solve that?
Well here is how I would go about it.

Since every parabola is symmetric about some line then it can not be injective,since it takes on every value twice,namely once before intersection with axis,and once after.Thus not injective.

Also from negative sign next to largest degree it is obvious that this function has a maxima.Thus any number larger then its maxima is not value of the function.Thus for some real number large enough there is no x such that it is equal to f(x).

To determine the image of the function it is only necessary to determine the maxima of the function(let it be called t from now on),then since function is not bound below then its image would be set off all numbers less than or equal to its maxima.

For D the largest injective domain would be either all x up to maximum point,or all x after maximum point,and the range would be still the image of f under R.This comes due to symmetry.
Find a counterexample to prove that the function $f(x)$ is not an injective nor surjective function. 


Say we have $-1.5,5 \in \mathbb{R}$, then $f(-1.5)= f(5) \in \mathbb{R}$, but $-1.5 \neq 5$, hence it is not injective.
Say we have $22 \in R$. There is no $x \in \mathbb{R}$ such that $-2x^2+7x+15 = 22$, hence it is not surjective.

Let the range be $Y:=\{y \in \mathbb{R},\;  y=-2x^2 + 7x +15\}$. Intuitively, what are all possible values of $f(x)$ for all $x \in \mathbb{R}$?