2026 AP Precalculus Free Response #1

Added: 2026-05-15

[00:00:00]All right, so we're looking at the 2026 AP precalculus FRQs. Um so we have a function here and this is the calculator portion. So we have a function G given by this and we have a graph of F up here. Okay, so makes sense. The function H is defined as G of F of X or this. I prefer this notation FYI, but that's uh they're showing you both. Oops. Uh find the value of H of 2 as a decimal approximation or indicate that is not defined. Show your work that leads to your answer. Okay, so H of 2 is basically plugging two into this guy right here. That's G of F of 2. Now, F of 2 is the Y value of this graph, right? You go to two here and you look at the Y value and say like, oh, this is the point 2 5. What 5 is the Y value, so we replace the F of 2 with 5, so it becomes G of 5. And G of 5 is then we got to take that 5, plug it in for X into there. And this is the case where we are going to use Desmos for this. Um so let me pull up Desmos over here. Let's just go ahead and write the expression. I like to use the expression here -4.792 + ln of 6x - 6.

[00:01:07]And I like to do that because um I can just visually see that it's I typed it in correctly. It's just less error prone. And then we just want to do G of 5 here.

[00:01:16]So then that's going to give me the number uh -1.614.

[00:01:21]Like that.

[00:01:23]Find all values for X where f of x is equal to 3 or indicate that there are no such value. So here we're going to say like the number the value of the function is always the Y value on the graph here. So then we're going to say like 3, where is the Y value equal to 3?

[00:01:37]There's only a single point here and the X value is 1. So that's X is equal to 1 there.

[00:01:43]Okay. Find all values of X for which g of x is equal to -1.5 or indicate there's no such value. So here what we're going to do is we're just going to graph it y = -1.5. Do that horizontal line just kind of like I did here, and we want to find the intersection. So, we're just going to say y = 1.5, and let's find that intersection point there. That's going to occur um x equal to 5.483.

[00:02:12]Like that. Okay?

[00:02:13]Determine the behavior of g as x value as x values increase decrease and get arbitrary close to 1. Express your answer doing the mathematical notation of a limit. So, we're going to be approaching 1 from the right side. So, we're going to get smaller. So, we're going to do the limit as x is approaching 1 from the right.

[00:02:33]So, cuz we're going to follow it this direction cuz there's the value of the function doesn't exist over here.

[00:02:38]There's no values for the graph, right?

[00:02:40]So, we're going to come from the left of g of x.

[00:02:43]And what do we want to say is And and the the value of the limit is what the y value is going to be doing. So, the y value is just getting more and more negative here. So, that approaches negative infinity. So, that would be the answer there.

[00:02:56]Is the function f invertible? Okay, invertible means that for every y value, there's a unique x value associated with it, right? That means another way to look at it is it's the passing the horizontal line test. That's what I was When I was learning math, that's what I was taught, but fundamentally, invertible means that there's only for every y value, there is a unique x value that corresponds to it. So, what we're going to say here is it is invertible invertible cuz no matter what whatever y value I decide to pick, there's only a single unique x value associated with it. That's what makes it invertible, okay? That there's a unique x value. So, yes, because there is a unique This is how I would write it. x value. I don't actually remember the rubrics x value for any y value in the y value in and the range.

[00:03:55]Okay.

[00:03:56]Give a reason for your answer part C based on the properties of the function F. Refer to points on the graph of F in your reasoning. Um I think this reasoning is fine or at least this is the reasoning I would go with. Let me double check real quick.

[00:04:08]I'm going to guess that this is a correct reasoning based off of it.

[00:04:10]There's no examples from past exams I can say, but like this is what you're trying to infer basically. I don't know if they need to be more specific than this, but basically for every unique Y value there is every Y value there is a unique only a single X value. They just want you to understand that that is how you define an inverse.