Chapter 3.3, Problem 47PS

Solution

(a)

The value of $f\left(x\right),f\left(f\left(x\right)\right),f\left(f\left(f\left(x\right)\right)\right)$ and $f\left(f\left(f\left(f\left(x\right)\right)\right)\right)$ when $n=2,\text{}\text{ and x=1}$ .

$f\left(x\right)=\frac{3}{2},f\left(f\left(x\right)\right)=\frac{17}{12},f\left(f\left(f\left(x\right)\right)\right)=\frac{577}{408}$ and $f\left(f\left(f\left(f\left(x\right)\right)\right)\right)=\frac{665857}{470832}$

Given:

The function to approximate the square root of a number n is $f\left(x\right)=\frac{x+\frac{n}{x}}{2}$ .

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to calculate $f\left(x\right)$ , substitute $n=2,\text{}\text{ and x=1}$ into the given function and simplify further as shown below,

  $\begin{array}{c}f\left(x\right)=\frac{x+\frac{n}{x}}{2}\\ =\frac{1+\frac{2}{1}}{2}\\ =\frac{1+2}{2}\\ =\frac{3}{2}\end{array}$

Now, to find the composition $f\left(f\left(x\right)\right)$ , replace x by the value of $f\left(x\right)=\frac{3}{2}$ in the function f as shown below

  $\begin{array}{c}f\left(f\left(x\right)\right)=\frac{\frac{3}{2}+\frac{2}{\frac{3}{2}}}{2}\\ =\frac{\frac{3}{2}+\frac{4}{3}}{2}\\ =\frac{\frac{9+8}{6}}{2}\\ =\frac{17}{12}\end{array}$

By following the same procedure, to find the composition $f\left(f\left(f\left(x\right)\right)\right)$ , replace x by the value of $f\left(f\left(x\right)\right)=\frac{17}{12}$ in the function f as shown below

  $\begin{array}{c}f\left(f\left(f\left(x\right)\right)\right)=\frac{\frac{17}{12}+\frac{2}{\frac{17}{12}}}{2}\\ =\frac{\frac{17}{12}+\frac{24}{17}}{2}\\ =\frac{\frac{289+288}{204}}{2}\\ =\frac{577}{408}\end{array}$

And finally to find the composition $f\left(f\left(f\left(f\left(x\right)\right)\right)\right)$ , replace x by the value of $f\left(f\left(f\left(x\right)\right)\right)=\frac{577}{408}$ in the function f as shown below

  $\begin{array}{c}f\left(f\left(f\left(f\left(x\right)\right)\right)\right)=\frac{\frac{577}{408}+\frac{2}{\frac{577}{408}}}{2}\\ =\frac{\frac{577}{408}+\frac{816}{577}}{2}\\ =\frac{\frac{332929+332928}{235416}}{2}\\ =\frac{665857}{470832}\end{array}$

Therefore, the values of the compositions are $f\left(x\right)=\frac{3}{2},f\left(f\left(x\right)\right)=\frac{17}{12},f\left(f\left(f\left(x\right)\right)\right)=\frac{577}{408}$ and $f\left(f\left(f\left(f\left(x\right)\right)\right)\right)=\frac{665857}{470832}$ .

(b)

How many times do you nee to compose the function in order for the result to approximate $\sqrt{2}$ to three decimal places? Six decimal places?

2 times, 3 times

Given:

The function to approximate the square root of a number n is $f\left(x\right)=\frac{x+\frac{n}{x}}{2}$ .

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

Note that the value of $\sqrt{2}\approx 1.414235$ .

Also note that the approximate values of the function and compositions obtained in above part are,

  $\begin{array}{c}f\left(x\right)=\frac{3}{2}\approx 1.5\\ f\left(f\left(x\right)\right)=\frac{17}{12}\approx 1.416667\\ f\left(f\left(f\left(x\right)\right)\right)=\frac{577}{408}\approx 1.41421568\\ f\left(f\left(f\left(f\left(x\right)\right)\right)\right)=\frac{665857}{470832}\approx 1.41421356\end{array}$

Therefore, the value of $\sqrt{2}$ up to three decimal places can be obtained by composing the function 2 times.