Question Video: Finding an Algebraic Expression for a Composite Function | Nagwa Question Video: Finding an Algebraic Expression for a Composite Function | Nagwa

Question Video: Finding an Algebraic Expression for a Composite Function Mathematics

Set 𝑓^(2) = 𝑓 ∘ 𝑓, 𝑓^(3) = 𝑓 ∘ 𝑓 ∘ 𝑓, and so on such that 𝑓^(2) 𝑥 = 𝑓(𝑓(𝑥)) and 𝑓^(3) (𝑥) = 𝑓(𝑓(𝑓(𝑥))), and so on. Suppose 𝑔(𝑥) = 4𝑥 − 5. Find 𝑔^(4) (3).

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Video Transcript

Set 𝑓 superscript two equal to 𝑓 of 𝑓, 𝑓 superscript three equal to 𝑓 of 𝑓 of 𝑓, and so on such that 𝑓 superscript two of 𝑥 equals 𝑓 of 𝑓 of 𝑥 and 𝑓 superscript three of 𝑥 equals 𝑓 of 𝑓 of 𝑓 of 𝑥, and so on. Suppose 𝑔 of 𝑥 equals four 𝑥 minus five. Find 𝑔 superscript four of three.

Let’s just get to grips with what the notation in this question is telling us. These superscripts denote the number of times that we’re applying a given function. 𝑓 superscript two of 𝑥 means 𝑓 of 𝑓 of 𝑥. So we take an input value 𝑥, apply the function 𝑓, and then apply the function 𝑓 a second time. We’re asked to find 𝑔 superscript four of three, so this means we take an input value of three and apply the function 𝑔 four times, each time to the result of the previous iteration.

Let’s see what this looks like then. 𝑔 of 𝑥 is the function four 𝑥 minus five. We take an input value, multiply it by four, and then subtract five. So 𝑔 of three is four times three minus five. That’s 12 minus five, which is equal to seven. Next, We need to find 𝑔 superscript two of three, which means 𝑔 of 𝑔 of three. We’re applying the function 𝑔 to the value of 𝑔 of three. Now we’ve just worked out that 𝑔 of three is equal to seven. So we’re taking the value seven as the input value to the function 𝑔.

We take this input, multiply it by four, and subtract five. That’s the 28 minus five, which is equal to 23. We’ve now applied the function 𝑔 twice. We need to apply it two more times. 𝑔 superscript three of three is 𝑔 of 𝑔 of 𝑔 of three. We’ve just worked out that 𝑔 of 𝑔 of three, or 𝑔 superscript two of three, is equal to 23. So replacing 𝑔 of 𝑔 of three with 23, we have 𝑔 of 23. And we see again that we are applying the function 𝑔 to the previous output. We multiply 23 by four and subtract five, giving 87.

We’ve now applied the function 𝑔 three times. We need to apply it once more. 𝑔 superscript four of three is 𝑔 of 𝑔 of 𝑔 of 𝑔 of three. We’ve just worked out that 𝑔 of 𝑔 of 𝑔 of three or 𝑔 superscript three of three is 87. And so we can replace this part of the function, giving just 𝑔 of 87. Once again, we’re taking the previous output value as the input to the function 𝑔. We multiply this value by four and subtract five to give 343. So we found then that 𝑔 superscript four of three, that’s the function 𝑔 applied four times to the input value of three, is 343.

Notice that the superscript has been written in brackets and that’s important because it distinguishes it from powers of 𝑔. 𝑔 and then a superscript four without brackets could mean that the function 𝑔 multiplied together four times or the function 𝑔 to the power of four, so four 𝑥 minus five to the power of four, which is not the same as applying the function 𝑔 four times each time to the output of the previous situation.

Our answer 𝑔 superscript four of three is 343.

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