Chapter 1.8, Problem 64E

(a)

To determine

To find: The composite function that gives the youngest sibling’s age in terms of the oldest and explain the way to obtain the answer.

Answer to Problem 64E

The composite function of the age is $\left(z\circ y\right)\left(x\right)=x-12$ this is obtained when the age relation for the siblings is $x\left(y\right)=2y$ and $y\left(z\right)=6+\frac{z}{2}$ .

Explanation of Solution

Consider the three siblings are $x,y,z$ where the age relation is,

  $x>y>z$

The age of oldest is twice the age of the middle sibling then,

  $x\left(y\right)=2y$

The middle sibling is 6 years older than the half of the youngest one then,

  $y\left(z\right)=6+\frac{z}{2}$

Then,

  $z\left(y\right)=2\left(y-6\right)$

Then, the composite function that represents the age of the youngest one in terms of the oldest one is,

  $\begin{array}{c}\left(z\circ y\right)\left(x\right)=z\left(y\left(x\right)\right)\\ =2\left(y\left(x\right)-6\right)\\ =2\left(\frac{x}{2}-6\right)\\ =x-12\end{array}$

(b)

To determine

To find: The age of the other two siblings when the age of the youngest sibling is $2$ years.

Answer to Problem 64E

The age of the middle sibling is $7\text{years}$ and the oldest sibling is $14\text{years}$ .

Explanation of Solution

Given:

The age of $z$ is 2 years.

Calculation:

Consider the expression for the age of the middle sibling is,

  $y=6+\frac{z}{2}$

Then.

  $\begin{array}{c}y=6+\frac{2}{2}\\ =7\end{array}$

The age of the oldest sibling is,

  $x=2y$

Then,

  $\begin{array}{c}x=2\left(7\right)\\ =14\end{array}$

Thus, the age of the middle sibling is $7\text{years}$ and the oldest sibling is $14\text{years}$ .