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Question Video: Composition of Three Functions Mathematics • 10th Grade

Given 𝑓(π‘₯) = π‘₯ βˆ’ 4, 𝑔(π‘₯) = π‘₯Β², and β„Ž(π‘₯) = 2π‘₯, evaluate 𝑓(𝑔(β„Ž(3))).

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Video Transcript

Given 𝑓 of π‘₯ equals π‘₯ minus four, 𝑔 of π‘₯ equals π‘₯ squared, and β„Ž of π‘₯ equals two π‘₯, evaluate 𝑓 of 𝑔 of β„Ž of three.

In this question, three functions, 𝑓, 𝑔, and β„Ž of π‘₯, have been defined. We’ve been asked to evaluate 𝑓 of 𝑔 of β„Ž of three. But what does this mean? Well, 𝑓 of 𝑔 of β„Ž of π‘₯ is a composite function. When we write composite functions like this, we work from right to left, from the function closest to the input backwards. So, this notation is asking us to take an input value of three, apply the function β„Ž to it, then apply the function 𝑔 to the result, and finally apply the function 𝑓 to the result of this. We can do this step by step.

Let’s begin then by evaluating β„Ž of three. β„Ž of π‘₯ is the function two π‘₯; it takes an input value and multiply it by two. So, β„Ž of three is two times three, which is equal to six. Next, we’re going to apply the function 𝑔. But we’re going to apply it to the value of β„Ž of three. β„Ž of three was six. So, 𝑔 of β„Ž of three is the same as 𝑔 of six. Now, 𝑔 of π‘₯ is the function which takes an input value and squares it. So, 𝑔 of six is six squared, which is 36.

We’ve now found then that 𝑔 of β„Ž of three is 36. And we just need to apply the function 𝑓. So, applying 𝑓 to this value of 𝑔 of β„Ž of three gives 𝑓 of 36. 𝑓 is the function which takes its input value and subtracts four. So, 𝑓 of 36 is 36 minus four, which is 32. So, by taking the input value three, applying the function β„Ž, then the function 𝑔, and then the function 𝑓, we found that 𝑓 of 𝑔 of β„Ž of three for the three given functions is 32.

It’s worth mentioning that we could also have taken a different approach to this question. We could have found an algebraic expression for the composite function 𝑓 of 𝑔 of β„Ž of π‘₯ by composing the three functions algebraically. And we could then have substituted π‘₯ equals three at the end and evaluated this. But this would’ve been more complicated, and there would’ve been more room for error than the method we chose, which was to substitute π‘₯ equals three at the beginning and then apply each function. Once again, we found that 𝑓 of 𝑔 of β„Ž of three is equal to 32.

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