Prove that the function g R 4 infinity defined by gx x4 4 i

Prove that the function g: R-> [ 4, +infinity) defined by g(x) =x^4 +4 is onto.

Solution

g(x) = x^4 + 4

y = x^4 + 4

y - 4 = x^4

fourthroot(y - 4) = x

Let y be in the given range, R

Then (y - 4) and fourthroot(y - 4) are also real numbers

Now, f(4throot(y - 4)) = (fourthroot(y - 4))^4 + 4
= y - 4 + 4
= y

Hence if y is in R, there exists an x in R such that f(x) = y

And thus, proved that the given function is ONTO