2026 AP Precalculus Free Response #4

本站添加: 2026-05-15

[00:00:00]Okay, [snorts] let's look at the FRQ number four of the 2026 AP precalculus exam. So, the functions G and H are given by this. Solve for G of X equal to this. So, e to the 2x equals 1 over e to the 6 for the values of X in the domain of G. First of all, domain of G is any value. You can do exponential to any number.

[00:00:20]Um so, it's just e to the 6, not e to the So, that this this is this part there. And exponential rule makes it e to the -6.

[00:00:28]How can we do that? Well, once you make them match, you can take the natural log of both sides. Or, you can just recognize that exponents must have to match. So, you get 2x equal to -6. So, X is equal to -3. Divide both sides by 2.

[00:00:41]Solve for h of X equal to 3 3 is equal to log 2 of 5x in the domain. Domain of H is going to be X is greater than 0.

[00:00:51]Uh because in this case, you can't plug in negative like you can't plug in 0 or negative into logs.

[00:00:57]So, uh how do we get rid of logs? We put 2 as the base.

[00:01:00]2 cubed is 8. 2 log base 2, they're inverses, they cancel each other. It gives you 5x. So, X is going to be 8 over 5 like that.

[00:01:09]Okay, the functions J and K are given by this. Rewrite this in the form 7 to the X. So, this is about exponential rules.

[00:01:15]When you have the same base, you just add the exponent. So, it's 3x + 1 + X, which is 7 4x + 1. 3x + X.

[00:01:25]Okay. Rewrite this expression as a sin x appears exactly once and no other trig functions are involved.

[00:01:32]Um sin x So, uh first of all, sin of 2x is a trig identity. That's 2 sin x cos x.

[00:01:42]Okay. And then sec And then secant of x is 1 over cosine of x.

[00:01:48]So, then this will be You multiply these two is 2 sin x cos x times 1 over cos x.

[00:01:56]Those cancel. You get 2 sin x.

[00:01:59]Like that.

[00:02:01]Okay, the function m is given by this.

[00:02:02]Find all input values for m on the interval that yield an output value of 1. So, we want tan squared of 3x has to equal 1.

[00:02:10]Now, that means tangent be very careful could be 1, but it could also be -1. Right?

[00:02:23]So, we want to think on our unit circle where is tangent equal to 1 or -1? It's going to be here, here, here, and here.

[00:02:30]That's where the x and the y values, the magnitude is 1. This would be positive 1, positive 1, negative 1, negative 1 for tangent. It's at those pi over four marks. So, uh our solutions are going to be 3x.

[00:02:41]Combining these two, both possibilities, you're going to have pi over four, you're going to have 3 pi over four, you're going to have 5 pi over four, you're going to have 7 pi over four.

[00:02:51]Now, how far do we go? Because we're going to divide by three, we just need to go not go past pi over two. So, I'm just going to keep going until I get past it. So, if I keep going around, that's going to be uh what? 9 pi over four, um and then uh 11 pi over four, and then we'll do one more, 13 pi over four. So, then if I divide by three, x is going to be pi over 12, 3 pi over 12, which is pi over four, 5 pi over 12, 7 pi over 12.

[00:03:26]And and pi over two we're going to stop at is um 2 pi.

[00:03:31]Uh in terms of 12 is 6 pi over 12.

[00:03:35]Right? So, multiply uh six and six. Yeah, 6 pi over 12 is pi over two. So, notice that we have already exceeded it.

[00:03:44]So, we've already gone past. This is past 6 pi over 12. This is past pi over two. So, I I don't need to include any of those.

[00:03:50]So, let's just simplify pi over 12, pi over 4, cuz that cancels, and then 5 pi over 12.

[00:03:58]Those would be all the values in there, like that.