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Determine the equation of the inverse function.$$f(x)=-5 x^{2}+3, x \leq 0$$

$f^{-1}(x)=-\frac{\sqrt{15-5 x}}{5}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

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So if we want to find the inverse of this one thing, we should also first check is to see if this function is 1 to 1. Because if not, then we need to, like, restrict our domain. Um, so one way we can do that is just by taking the derivative and showing that this is always increasing or decreasing So f prime of X is going to be well, we use power rules. That would be negative 10 x here. But we have this restriction of X being less than or equal to zero. So if we come over here and multiply this by X, that tells us negative 10 X is going to be greater than or equal to zero, which again is f prime of X o f. Prime of X is always greater than or equal to zero, which implies f a bex is always increasing. So, uh, yes, Technically, we really didn't need to do this, but it's always good to just go ahead and do that just to make sure. So now, to find the inverse, we can go ahead and first replace FX with y and then Interchange X and y so this is going to be X is equal to negative five y squared, plus three. So let's first get y squared by itself. So I'm going to subtract three. So the X minus three, then we divide by negative five and this is why squared. So now we would take the square root on each side here, and this is going to be equal to the absolute value of why on the right side. So it's important that we have this year. So now this is supposed to be the range of our inverse. So if we know that the domain of our function is supposed to be negative infinity to zero, then remember, this means the range of our inverse is going to be the same. So it be negative infinity to zero. So when we look at this, the absolute value of why isn't just going to be why is actually going to be negative? Why, um so we'd end up where for this is equal to square root of X prize three over negative five and then we could just multiply the negative one over. So we end up with y is equal to, uh, negative square root of X minus three over negative five. And then I'm just going to distribute that negative to the top, to flip those around so it be square root or negative square root of three minus X, all over five. And then this is f interest. I mean, you can also leave it like how it was in the previous line, but this just looks a little bit prettier to me. And then just to make sure, um, it's clear this square root is for everything on the inside.

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