Determine whether the function is one-to-one. If it is, find...

Determine whether the function is one-to-one. If it is, find a formula for its inverse.\\ $f(x) = \frac{4}{3}x + 8$\ Select the correct choice below and, if necessary, fill in the answer box to complete your choice.\ A. The function is one-to-one. $f^{-1}(x) = $\ (Simplify your answer. Use integers or fractions for any numbers in the expression.)\ B. The function is not one-to-one. There is no inverse function.

Transcript

00:01 Hello, everybody.

00:02 So in this problem, the first thing we're asked to do is determine whether this function is one -to -one, because if we're going to have an inverse, we're going to find an inverse, and this function, first of all, has to be one -to -one.

00:12 So to do that, a one -to -one function means that for every input, so in this case, for every x value that we get, we have to have a different output.

00:21 So there's not two numbers for x that we can put in that will give me the same number for f of x or the same output.

00:28 But, well, let's just say we have really just three cases to consider here.

00:32 We have when a is zero, when a is positive, and when a is negative.

00:39 Well, if a is zero, then obviously that's just going to be f of zero.

00:46 And we're saying a is x here in this situation.

00:48 We're plugging in some number a for x.

00:50 When i plug in zero, that's going to be zero cubed minus five, which is negative five.

00:54 When a is a positive number, then i'm going to get whatever that number is cubed minus five.

01:06 So some positive number cubed minus five.

01:10 If a is less than zero, i put zero again.

01:16 If a is less than zero, then let's say we're using like negative a, then i'm going to get negative a cubed minus five.

01:24 And the important thing here is that anything negative cubed is just, just that negative times a cubed.

01:31 And so i never get the same answer, no matter what number i plug in.

01:35 So a different example would be if i use the function x squared.

01:39 So let's say, what are you doing with g of x? if i do negative one for that, well, negative one times negative one is still one.

01:49 If i do one for that and square that, that's also one.

01:54 So i got the same output for two different inputs, which means that's not a one.

01:59 One function.

02:00 So this was not a one to one function.

02:02 However, my function x cubed minus five is because no matter what i plug in, i'm going to get a different value every time.

02:08 The negatives are going to make me where i have a same output for two different x's.

02:14 So this is one to one.

02:16 So our first thing is one to one.

02:20 We're going to put a big old check.

02:21 That is a one to one function.

02:23 Now, because it's a one to one function, we can find it's inverse.

02:27 So we'll rewrite the function right here.

02:31 And then i'm actually going to write that this is y instead of f of x, so f of x and y, we can use intertrain everybody here...

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