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Determine the equation of the inverse function.$$f(x)=2 x^{2}+3, x \geq 0$$

$f^{-1}(x)=\frac{\sqrt{2 x-6}}{2}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

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Find the inverse function …

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Find the inverse of $f(x)=…

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So before we find an inverse, we should always check to make sure the function is 1 to 1. And if not, we need to restrict the domain. Uh, so here, if we were to go ahead and first just take derivative of this weekend, F Prime of X is equal to will use power rules. It would be four X, but we have this restriction of X being greater than equal zero. So now, if we go ahead and multiply this by four, we get four x greater than or equal to zero, which would then say F prime of X is greater than equal zero, which implies f of X is always increasing. Okay, uh, so now let's come down here and we can go ahead and actually find the emperor. So I guess this last part isn't really needed, but it's always good to check. Uh, so let's go ahead and replace. This will fly now, and an interchange x and Y So we have X is equal to, um, two y squared plus three. Then we need to go ahead and get y squared by itself. So it would be X minus 3/2 is a good Hawaii squared, and now we go ahead and take the square root here. So now this is supposed to be the absolute value of why now, to figure out if this is going to be negative, Why or just why we need to come up here and think, Well, what is supposed to be the, um the range of our inverse? Well, if we look at the domain of our function, it's X is greater than equal zero, which would imply that the inverse or the range of this is supposed to be just greater than equal to zero as well. Because, remember, we just switch these around, so that means this has to be greater than zero, which would just give us why. So that would tell us that this is going to be our verse here and again. It is important that you kind of make that distinguish mint, that this is an absolute value fly, and it just so happens in this case it is why

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