5

3. Graph f (*, =Vx+5 4and its inverse. [2]Define even and ccd svmmetry. Cetermlne Ifthe functlon / *)e3x *-3 Is even; ccdc; neither. Answers nct acccmpanied by work...

Question

3. Graph f (*, =Vx+5 4and its inverse. [2]Define even and ccd svmmetry. Cetermlne Ifthe functlon / *)e3x *-3 Is even; ccdc; neither. Answers nct acccmpanied by work wlll be considered Incomplete. [2]

3. Graph f (*, =Vx+5 4and its inverse. [2] Define even and ccd svmmetry. Cetermlne Ifthe functlon / *)e3x *-3 Is even; ccdc; neither. Answers nct acccmpanied by work wlll be considered Incomplete. [2]



Answers

Find the inverse function (on the given interval, if specified) and graph both $f$and $f^{-1}$ on the same set of axes. Check your work by looking for the required symmetry in the graphs.
$$f(x)=\sqrt{3-x}, \text { for } x \leq 3$$

Okay, so we want to graph the following equation. Let's notice that this here is a piece wise function. So it's not that let's start with our X is equal to two. When X is equal to two, we know that our Y value is equal to three. So we have one x addicted to Why is too or is three Now let's graft this equation when X is not equal to two. So lets you know that this is important for so we have an X intercept of negative too. And then we're going to go up three units 123 and to the right, one up 3123 into the right one. So it's not that instead of a solid circle, we're going to have an open circle here. Okay, so our graph is going to look like this and then extends upwards like so

This is a rash if lines are inverse function. So let's start by saying after Backstreet, that's why. And now let's into change your ex center way. Okay, I know it's all for what? So we'll square both sides we have export is you put three months. Why we'll add wise you both sides a knowledge Subtract X squared for most sites. So we get why is equal to negative X squared plus three. And this is equal to our inner assumption. And we also have to note that this is for X less than or equal to three correction, etc. Of. If we have X less than or equal to to be here, then we see that the square root of three minus let's say one that's equal to the square root of two. Okay, so we have X is one and F of X is the square root of two. And now let's check that with our inverse function. So if we have f inverse of our range, that's the square root of Chu. We get negative square root of two squared plus three that's equal to negative one or a night of to plus three, which is equal to one I know here that we check in values of X square root of two. Mitzi for over two is equal. Teoh 1.4 and that's still less than three. Yeah, we have four x less than or equal story.

The following problem. We want to find the inverse function. Um And graft both F and F inverse on the same axis. And we're going to check on our work. So we're gonna have F of X equaling negative expert minus four X -3. That's going to be when acts is less than or equal to. And then you have to uh huh. The way that we saw this is by completing the square, What this is going to look like is we let y equal this, then we add the three overs. That way we can complete the square. Um We also want to actually before we do that we're gonna divide out the negative value. So that's going to complex And we're gonna subtract three But this are gonna be a negative y. So negative I -3. Then this is going to become plus uh plus four with the put on both sides. So it becomes negative Y plus one. We complete the square. We solve for x twitter to white and that will be Are inverse function which is negative. Route 1 -1

This question gives you a function of which to find the inverse and also wants you to graph it. So we have f of X equals X plus three squared. Now, let's take a look at what this is gonna look like. I'm gonna draw the axes and we'll go from there. So we knew that the parent function of this is just X squared, which is just a Ceravolo like this. Oh, I should also mention that the domain of this function the domain is X is greater than or equal to negative three. The parent function is X squared here. Now we have a shift in it because we're adding three to the X term. That means we shifted three toe left. So I'm gonna erase the parent function and we will draw some tick marks and go from there. So if this is X equals negative three, it's negative to a negative one. When x is negative, three f of X will be zero when x is negative. Two f of X will be one when f of X is negative one or when X is negative. One f of X will be four salami draw awesome marks up here and so on like that. It will eventually get there, So yeah, this is what we have. And remember, it goes from negative three on, so let's try to find an inverse. But before we get it, we do that. We have to look at the range of this function. It's important that way. We know the range of the function because the range of the function becomes the domain of the inverse function. Because we have a restricted domain here. We knew that a range, uh, that that our range is also gonna be restricted. Since we've already drawn that graph, we don't really have to do much working out. We can see that. Um, from what we're given, it looks like the range starts at zero and goes up. In fact, there's never a X value that can give you a ffx less than zero. So the range is gonna be 02 infinity. Actually, that zero is inclusive, so it should be a square bracket. The range is zero to infinity going up like that, that's the range of our function which meets the domain of our inverse function. So we're only going to draw are inverse function from the Y axis over to the right. We'll get there. But first we have to figure out what this function actually is. So we know when we're doing, um, inverse functions. Our method is to just switch the places of X and y so inverse function is gonna look like this X equals why plus three squared. And now we have to do is solve. For what? Well, we know that we're gonna take the square root of both sides. It's X equals y plus three squared. Excuse me. Just wipe those three and to get why on its own will subtract three from both sides would get root. X equal minus three equals Why? And that is our f inverse of X. That's are inverse function. Let's go ahead and graft that. I'm gonna graph it in green just so it can tell difference. We know what our square root of X function looks like, but this one is shifted down. Three. So let's make three tick marks down this way. Um and we'll start here. We know that when X is one squared of X is one. So one month through will be negative, too. So we'll have f inverse of negative of one is negative. Two. When F is, let's say, four or acute when X is four because it's the next square number. When X is four f inverse of X will be squared a four, which is two minus three, which is negative one. So that will bring us here and we'll have a function that looks like this. It will go on like that. So that is our inverse function. And you can see that this is what we expect would expect the in perception to be the function reflected over this X equals y line. And so this is what your final answer will look.


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