Question

Find the range of the function.

f(x) = x2 + 3

Answer 1

Range:
`[3,\infty)`

Explanation:

We have been given a function
`f(x)=x^2+3`. We are asked to find the range of our given function.

We know that range of a function is values of dependent variable (y) for which function is defined.

We can see that our given parabola is in vertex form
`y=a(x-h)^2+K` as
`y=1(x-0)^2+3` with a vertex at point
`(0,3)`.

Since our given parabola is an upward opening parabola, so point
`(0,3)` is the minimum point.

We know that the range of an exponential function is form
`ax^2+bx+c` with a vertex at (h,k) is:

If
`a>0`, then range is
`f(x)\geq k`

If
`a<0`, then range is
`f(x)\leq k`

Since the value of a is positive and
`k=3`, therefore, the range of our given function would be
`f(x)\geq 3` that is
`[3,\infty)`.

Answer 2

To find the range, you need to locate the vertex:

equation for vertex = ax^2 + bx +c

in x^2 + 3, a = 1, b = 0 and c = 3

now vertex form = a(x+d)^2 +e

d = b/2a = 0/2*1 = 0

e = c-b^2/4a = 3-0^2/4*1 = 3

The equation becomes: (x+0)^2 +3.

The range is Y ≥ 3 ( greater than or equal to 3)