# Video: Finding the Composite Function of Two given Functions Then Evaluating It

Given that the function π(π₯) = 19π₯Β² and the function π(π₯) = β2π₯, determine (π β π)(π₯) in its simplest form, and evaluate (π β π)(1).

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### Video Transcript

Given that the function π π₯ equals 19π₯ squared and the function π π₯ equals negative two π₯, determine π circle π of π₯ in its simplest form and evaluate π circle π of one.

The first thing we notice about this question is this notation, and I called it π circle π. It can also be known as π composite π. And π composite π, well it helps to understand what it is cause itβs actually a composite function, so what does π circle π or π composite π actually mean?

Well π circle π is equal to πππ₯, and what this means is first you apply π and then you apply π to the result. So I can actually think of it in terms of a function machine, cause there we have our input, and then weβre gonna complete function π, which is gonna be 19 multiplied by our input all squared, and then weβre gonna take the value we get from that; that becomes our new input, and then weβre gonna complete our function π, which is negative two multiplied by that input, and then that will give us our final output.

So as you can see, π circle π, we now know what it is and we now know how to use it, so weβre gonna use it to solve this problem. So we have our two functions and actually if we think about it, our input is π₯, so if we complete the first function, so function π on π₯, weβre gonna get 19π₯ squared.

So now what we need to do is, to solve this and find out what π circle π is, is weβre gonna actually substitute in 19π₯ squared for π₯ in our function of ππ₯. So we can say that π circle π is equal to negative two multiplied by 19π₯ squared, which gives the final answer of π circle π is equal to negative 38π₯ squared.

Okay and we can check that using our function machine. So this here, our input, is π₯ and then itβll go through our first stage of our function machine, which would be 19 multiplied by π₯ squared, which gives us 19π₯ squared. And then we complete the final part, which would be π, so weβve got a new input that is negative two multiplied by the new input of 19π₯ squared, which gives us negative 38π₯ squared. Fantastic! This is what we need.

And so weβve now found π circle π. Having now found π circle π, we can actually now solve the second part of our problem, which is evaluating π circle π one. What this actually means is that we want to substitute π₯ equals one, so we want to make the value of π₯ equal to one in our composite function.

So this is gonna give us negative 38 multiplied by one squared, which gives us our answer of π circle π or π composite π of one, where π₯ is equal to one, is equal to negative 38.

So there we have it. Weβve solved both parts of the problem. So the key thing to remember from this question is that π circle π or π composite π means that weβre gonna do the π-function first and then the π-function upon it, and itβs the same as π of π of π₯.