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 π^(2) = π β π, π^(3) = π β π β π, and so on such that π^(2) π₯ = π(π(π₯)) and π^(3) (π₯) = π(π(π(π₯))), and so on. Suppose π(π₯) = 4π₯ β 5. Find π^(4) (3).

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### Video Transcript

Set π superscript two equal to π of π, π superscript three equal to π of π of π, and so on such that π superscript two of π₯ equals π of π of π₯ and π superscript three of π₯ equals π of π of π of π₯, and so on. Suppose π of π₯ equals four π₯ minus five. Find π superscript four of three.

Letβs just get to grips with what the notation in this question is telling us. These superscripts denote the number of times that weβre applying a given function. π superscript two of π₯ means π of π of π₯. So we take an input value π₯, apply the function π, and then apply the function π a second time. Weβre asked to find π superscript four of three, so this means we take an input value of three and apply the function π four times, each time to the result of the previous iteration.

Letβs see what this looks like then. π of π₯ is the function four π₯ minus five. We take an input value, multiply it by four, and then subtract five. So π of three is four times three minus five. Thatβs 12 minus five, which is equal to seven. Next, We need to find π superscript two of three, which means π of π of three. Weβre applying the function π to the value of π of three. Now weβve just worked out that π of three is equal to seven. So weβre taking the value seven as the input value to the function π.

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

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

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

Our answer π superscript four of three is 343.