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If $f(x)=\frac{2 x-3}{3 x+4},$ find (a) $f^{-1}(-3),$ (b) $f^{-1}(3)$

(a) $-9 / 11$(b) $-15 / 7$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

McMaster University

University of Michigan - Ann Arbor

Lectures

02:33

If $f(x)=\frac{2 x+1}{3 x+…

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If $f(x)=x^{2}+1,$ find

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If $f(x)=\sqrt{3 x+2},$ fi…

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If $f(x)=-x^{3}-x^{2}+3,$ …

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00:53

Find $f$ . $f^{\prime}…

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If $f(x)=x^{2}+4 x+3$, fin…

01:02

If $f(x)=\frac{1}{2} x-\fr…

so before we actually try to figure out what the inverse of this is going to be. One thing we can do is try to figure out what is the range of f going to be to ensure that the inverse is defined at negative 33 Because sometimes when we saw for the inverse, um, we end up losing some information and numbers that we think we might be able to plug in. We actually won't be able to, um so first the range about And if you were to kind of, like work this through, um would end up being just negative infinity to two thirds union, two thirds to infinity because it's just some rational functions we have. No, it has horizontal attitude at two thirds. And then nothing really else anywhere else. Um, so that means the domain of F inverse is just the range of our original. And so you can see here how both of these negative three and three will be in its domain, so we can just go ahead, replace this with or why now and then interchange the access and try to solve for y so X is equal to two y minus 3/3 y plus four. So first, um, we're just gonna clear the denominators, so it would be three x y plus four x is equal to two y minus three. Now we want to get the wise on one side and everything else on the other. So the three XY minus two y and I'm going to subtract the four X over him so minus three minus four X, then we can go ahead and factor of why. So why? Times three X minus two is equal to, and I'm going to factor the negative out here, um, Soviet three X plus four X and I'm going to divide that over. So we end up with y is equal to negative three plus four X all over three X minus two and then we can take this negative distributed to the denominator, and that would be three plus four X is equal to two minus three acts. So this is out verse. So let me go ahead and pick this up and scoot it up here. Then we can just plug negative three and three into here. So, uh, this would be three plus the negative 12 all over two plus nine. Ah, so that would be negative. 9/11. So that is our solution for a and then for B, this is going to be three plus 12 over, um, two minus nine, which is going to be 15 over seven, or it would be negative seven. So just being negative, 15 sevens, and that would be our solution for beef.

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