# How do you find the domain and range of f(x)=x^2 + 3?

Jan 8, 2018

color(red)("Domain" : x in RR

$\textcolor{red}{\text{Range} : y \ge 3}$

#### Explanation:

The first thing we can do is sketch the function, ${x}^{2} + 3$ is just the simple graph ${x}^{2}$ shifted 3 upward, using our knowledge of trnsformations...

${x}^{2}$:

graph{x^2 [-10, 10, -5, 5]}

${x}^{2} + 3$:

graph{x^2+3 [-9.375, 10.625, -1.56, 8.44]}

Hence to find the domain we must consider what values of $x$ will give out a real number of $y$, what values the graph is valid for, looking at the graph, we see that all values of $x$ gives a value for $y$, hence:

color(red)("Domain" : x in RR

As we know ${x}^{2}$ gives a vlaue for all values, possitive and negative

Now to consider the range, we just nned to aks what are all the values that $y$ can take on, we see the smallest value of  is  y = 3$, \mathmr{and} a l l t h e o t h e r v a l u e s o f$y above also can be found..

$\textcolor{red}{\text{Range} : y \ge 3}$