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Chapter 1: Problem 54

If \(\mathrm{f}(\mathrm{x})=[\mathrm{x} /(\mathrm{x}-1)], \mathrm{x} \neq 1\) then \(\left(\right.\) fofof \(\ldots \mathrm{f}_{(17 \text { times })}(\mathrm{x})\) is equal to \(\ldots \ldots\) (a) \([x /(x-1)]\) (b) \(x\) (c) \([x /(x-1)]^{17}\) (d) \([17 \mathrm{x} /(\mathrm{x}-1)]\)

Short Answer

Expert verified
The result of the function composition f( f( f( ... f(x)... ) ) ) 17 times is (a) \(\frac{x}{x-1}\).

Step by step solution

01

Analyze the given function f(x)

The given function is f(x) = [x/(x-1)], x ≠ 1.
02

Find f(f(x))

To find the composition of f with itself, i.e., f(f(x)), substitute f(x) in place of x in the given function: f(f(x)) = f\(\left(\frac{x}{x-1}\right)\) = \(\frac{\frac{x}{x - 1}}{\frac{x}{x - 1} - 1}\) Now simplify the expression: = \(\frac{\frac{x}{x-1}}{\frac{x}{x-1} - \frac{x-1}{x-1}}\) = \(\frac{\frac{x}{x-1}}{\frac{1}{x-1}}\) This simplifies to: f(f(x)) = x
03

Find f(f(f(x)))

Since we found that f(f(x)) = x, therefore f(f(f(x))) = f(x) which is \(\frac{x}{x-1}\).
04

Identify the pattern

Notice that after two function compositions, we return to the initial function f(x). To generalize for 17 compositions, we can divide 17 by 2 to determine if the result will be f(x) or x.
05

Calculate the result for 17 compositions

We know that 17 compositions can be represented as: f( f( f( ... f(x)... ) ) ) Divide 17 by 2: 17 ÷ 2 = 8 with a remainder of 1 As we have a remainder of 1, this indicates that the final result will be one f(x) composition after reaching x which means the result will be f(x) itself.
06

Conclusion

Therefore, the result of the function composition f( f( f( ... f(x)... ) ) ) 17 times will be equal to: f(x) = \(\frac{x}{x-1}\) So, the correct answer is (a) \(\frac{x}{x-1}\).

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Most popular questions from this chapter

Suppose sets \(\mathrm{A}_{\mathrm{i}}(\mathrm{i}=1,2, \ldots 60)\) each set having 12 elements and set \(\mathrm{B}_{\mathrm{j}}(\mathrm{j}=1,2,3 \ldots . \mathrm{n})\) each set having 4 elements let \({ }^{60} \mathrm{U}_{\mathrm{i}=1} \mathrm{~A}_{1}={ }^{\mathrm{n}} \mathrm{U}_{\mathrm{j}=1} \mathrm{~B}_{\mathrm{j}}=\mathrm{C}\) and each element of \(\mathrm{C}\) belongs to exactly 20 of \(\mathrm{A}_{i}\) 's exactly 18 of \(\mathrm{B}_{j}\) 's then \(\mathrm{n}\) is equal to (a) 162 (b) 36 (c) 60 (d) 120

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