GCSE Maths "Best Guess" Paper 1 Higher Topic Focus: Q20 Functions & Completing the Square | 14th May

Añadido: 2026-05-14

[00:00:11][music] >> Okay, so we're looking at functions and completed square form. Going to look at this in two parts cuz it's quite a large question.

[00:00:25]So we're going to start by having a look at the functions here where we have a substitution in part A and an inverse function in part B.

[00:00:33]Now it says here the f of x function is 3 over 4x - 5 where x cannot equal 5 over 4. Now it can't equal 5 over 4 because if you substituted 5 over 4 into that denominator, you'd get 0 on the bottom and you're not allowed to divide by 0. So we can't use the x value 5 over 4, but we can use anything else.

[00:00:54]Now this particular question has said to use the x value of 1/4. So it wants us to substitute 1/4 into that denominator.

[00:01:03]Now rather than writing this as one big messy fraction, I'm just going to substitute it into the bottom. So 4 * 1/4 - 5.

[00:01:13]4 * 1/4 is 1/4 of 4, so that's 1. So it's 1 - 5 which equals -4.

[00:01:21]So when I substitute -4 now into the bottom, we get 3 over -4 or 3/4. Or of course, you could write that as a decimal, -0.75.

[00:01:36]There we go. There is my answer, -3/4.

[00:01:40]Part B is an inverse function. Again, different ways of approaching an inverse function. I like to rewrite the function, so f of x equals 3 over 4x - 5 as y equals 3 over 4x - 5.

[00:01:59]So when we're doing the inverse function, we are finding the function that does the reverse of this one, which essentially means I just need to in this particular formula that I now have written, I need to make X the subject.

[00:02:12]To make X the subject here, a couple of different ways that we could do that, but ultimately, I need to move that denominator to the other side.

[00:02:20]So, I need to multiply by 4x minus five.

[00:02:27]Now, when we do that, we get Y lots of 4x minus five is equal to three.

[00:02:37]Remember, I'm trying to make X be the subject here.

[00:02:40]Now, there are a few different ways that I could go next. I could divide by Y, and then I could add the five over, divide by four, and take a few steps.

[00:02:48]I don't think it's going to look very nice, though, and typically, I would always expand the bracket. So, I'm going to expand the bracket, see what happens.

[00:02:55]We get when we multiply this bracket out, we get 4xy minus 5y.

[00:03:06]And that equals three.

[00:03:09]This should work quite nicely.

[00:03:11]I'm going to run out of space, so I'll come up here.

[00:03:14]Next step is I need to isolate the X.

[00:03:16]So, we're going to add 5y to the other side.

[00:03:20]So, that now leaves me with 4xy equals three plus 5y. And keep remembering, it's X that we're trying to make the subject here.

[00:03:31]So, I can actually finish this off in one step. Now, although I've written it in alphabetical order, I could write 4xy as 4yx, because I just need to divide by the 4y.

[00:03:42]And that will leave X on its own, making X the subject.

[00:03:45]So, I'm going to divide by 4y, and that leaves me as x equals 3 + 5 y over 4 y.

[00:04:00]Now to finish this as an inverse function, typically on the exam it will say f minus one of x equals and you just need to write the function on that dotted line.

[00:04:11]When we write the function, these y's that we put there were just to help us rearrange the formula so that we didn't get f of x confused with the letter x.

[00:04:21]It's a tough enough question without having to differentiate between different x's.

[00:04:26]So when we do that, it becomes 3 + 5 x over 4 x replacing those y's back with x's and there is our inverse function.

[00:04:39]So there we go. We are changing the subject within that with, you know, not the nicest of formulas to actually have to rearrange.

[00:04:47]And there we go. That was substitution into a function and finding an inverse function.

[00:04:54]Now the part c to this question says the function g is such that g of x equals 2 x squared minus 8 x minus 5.

[00:05:02]Express g of x in the form and then we have a bracket x minus b squared plus c where a, b and c are integers.

[00:05:13]So that is asking us to write it in completed square form.

[00:05:17]Which isn't normally something we would see within a function. This is a nice interesting question.

[00:05:24]Now it does have a coefficient of x squared greater than 1.

[00:05:29]Which means before we complete the square, technically it's part of completing the square, but before we complete the square like we would normally do if it was a coefficient of 1, we're going to divide everything by 2.

[00:05:41]Now that means factorizing by 2 cuz we're not changing the expression. We are just getting the expression in the form x squared.

[00:05:49]So, x squared minus 4x minus not very nice cuz we have an odd number.

[00:05:56]I'm going to leave it as 5/2 for the moment rather than writing it as 2.5. We can change it if we need to.

[00:06:05]Now, I need to complete the square for what's inside the bracket.

[00:06:09]Now, it's up to you. What you could do is just rewrite this to the side.

[00:06:14]Because when we complete the square, we introduce another bracket. So, it'll start to get very messy if it's a bracket inside a bracket.

[00:06:21]So, let's just complete the square from here. So, we halve the coefficient of x, which makes it x minus 2 squared.

[00:06:30]Then, if we expand this bracket, we would get plus 4. However, we want minus 5/2 or I think at this point minus 2.5.

[00:06:43]So, to get from 4 to minus 2.5, we minus 6.5.

[00:06:48]It's actually much easier using a decimal there even though it's non-calculator.

[00:06:53]So, that is now in completed square form.

[00:06:56]However, that needs to go back in the bracket.

[00:06:59]So, it's two lots of in brackets >> [snorts] >> bracket x minus 2 squared minus 6.5 close brackets.

[00:07:11]Typically, when we have two sets of brackets like this, these outer brackets I would write using square brackets. You don't have to, but that's typically how I'll write it so I can differentiate between the brackets.

[00:07:22]Now, I do need to multiply that two back in.

[00:07:25]But thankfully, the two just slots at the start of this first bracket. Don't have to expand the double bracket and times it all by two.

[00:07:32]It's just that 6.5 I need to times by two and that's not too bad. It's going to be minus 6 * 2 is 12. 0.5 * 2 is 1. So, 13.

[00:07:43]That is now written in the form that it asked for. A is two, B is also two, cuz it already had the negative there, and c is -13. Had a plus there in that form that it has to be written in, but that's okay. Just means the value of c is -13.

[00:08:01]So, there we go. That is writing a function in completed square form where the coefficient of x was bigger than one. So, quite a challenging one there, but there we go. Hopefully some good practice on completing the square.