# I need to prove that the following is not surjective. how do i do

1. May 13, 2010

### donmkeys

i need to prove that the following is not surjective. how do i do that?

let f:R->R be the function defined by f(x)=x^2 + 3x +4.

2. May 13, 2010

### sephiseraph

Re: surjective

You need to find at least one point in the codomain for which there is no mapping from the domain through f.

In other words, that function can only be surjective if we can choose some x to make f(x) equal to any value we like in R.

Clearly this function is bounded below by its minimum at x = -1.5,
which is f(-1.5) = 2.25 -4.5 + 4 = 1.75.

Observing that f has a minimum value of 1.75, we see that it is impossible to find any x in the real numbers such that f(x) is less than 1.75, thus f is not a surjective function.

3. May 14, 2010

### HallsofIvy

Re: surjective

The simplest thing to do is to note that the solutions to the equation $x^2+ 3x+ 4= 0$ are given by
$$\frac{-3\pm\sqrt{9- 4(1)(4)}}{2}= \frac{-3\pm\sqrt{-7}}{2}$$
and so are not real numbers.