Question 316861
Simplest way to figure this out is to graph it.


The graph would look like this:


{{{graph(600,600,-10,10,-30,30,x^2 + 3)}}}


You can see that, for all values of x, not just positive values of x, there is only one value of y for each value of x.


This makes the equation a function.


In this graph, y = f(x) = x^2 + 3


To find the inverse of the function, you would replace y with x and x with y and then solve for y.


Your equation is y = x^2 + 3
replace y with x and x with y to get:
x = y^2 + 3
solve for y to get:
y = +/- sqrt(x-3)


A graph of the inverse equation would look like this:


{{{graph(600,600,-30,30,-10,10,sqrt(x-3),-sqrt(x-3))}}}


You know this is an inverse equation, because it reverses the operation.


The inverse equation only exists when x >= 0.


When x = 5, the original equation gets you 5^2 + 3 = 25 + 3 = 28


In the inverse equation, when x = 28, the inverse equation gets you sqrt(28-3) = sqrt(25) = 5.


With the original equation you go from 5 to 28.
With the inverse equation you go from 28 to 5.


While your original equation is a function, your inverse equation is not.


In a function, every value of x must yield only 1 value of y.


If any 1 value of x yields more than 1 value of y, then the equation is not a function.


In that case, it is called a relation.