Chapter 3, Problem 9TP

Solution

The value of $g\left(f\left(9\right)\right)$ .

  $108$

Given:

The functions $f\left(x\right)=2x^{\frac{1}{2}}\text{ and }g\left(x\right)=3x^2$ .

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: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• Sum of like radicals’ property: $p\sqrt[n]{a}+q\sqrt[n]{a}=\left(p+q\right)\sqrt[n]{a}$
• 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 find the value of $g\left(f\left(9\right)\right)$ , first find the value of $f\left(9\right)$ by substituting 9 for x int the function f and simplify further as shown below

  $\begin{array}{c}f\left(9\right)=2\cdot {\left(9\right)}^{\frac{1}{2}}\\ =2\cdot {\left(3^2\right)}^{\frac{1}{2}}\\ =2\cdot 3^{2\cdot \frac{1}{2}}\\ =2\cdot 3^1\\ =6\end{array}$

Now, to find the value of composition $g\left(f\left(9\right)\right)$ , replace x by the value of $f\left(9\right)$ into the function $g\left(x\right)$ and simplify further as shown below

  $\begin{array}{c}g\left(f\left(9\right)\right)=g\left(6\right)\\ =3\cdot {\left(6\right)}^2\\ =3\cdot 36\\ =108\end{array}$

Thus, the value of the composition is $108$ .