Chapter 1, Problem 5RCC

(a)..

To determine

To find: The graph of the equation of the form $y=x^{p/q}$ .

Explanation of Solution

Given:

  $y=\sqrt[3]{x},y=\sqrt{x},y=x,y=x^2,y=x^3\text{on}\text{the}\text{interval}0\le x\le 2$

Concept used:

Let $n$ be a positive integer greater than 1 and let a be a real number

If $a=0,\text{then}\sqrt[n]{a}=0$

If $a>0,$ then $\sqrt[n]{a}$ is the positive real number b such that $b^n=a$

Calculation:

The equation of the form $y=x^{p/q}$

Where $p$ and $q$ are positive integers and $x\ge 0$

  $y=\sqrt[3]{x}$

  $y={\left(x\right)}^{1/3}$

Comparing the equation $y=x^{p/q}$

Where $p=1,q=3$ ( $p$ and $q$ are positive integers)

  $y=\sqrt{x}$

  $y={\left(x\right)}^{1/2}$

Comparing the equation $y=x^{p/q}$

Where $p=1,q=2$ ( $p$ and $q$ are positive integers)

  $y=x$

  $y={\left(x\right)}^{1/1}$

Comparing the equation $y=x^{p/q}$

Where $p=1,q=1$ ( $p$ and $q$ are positive integers)

  $y=x^2$

  $y={\left(x\right)}^{2/1}$

Comparing the equation $y=x^{p/q}$

Where $p=2,q=1$ ( $p$ and $q$ are positive integers)

  $y=x^3$

  $y={\left(x\right)}^{3/1}$

Comparing the equation $y=x^{p/q}$

Where $p=3,q=1$ ( $p$ and $q$ are positive integers)..

(b)

To determine

To find:

The set of all point which is lie in the $\left(x,y\right)$ in a coordinate plane such that $y/x<0$ .

Answer to Problem 5RCC

The point $\left(x,y\right)$ in a coordinate plane such that $y/x<0$ Belong to the second or fourth quadrants.

Explanation of Solution

Given:

The point $\left(x,y\right)$ in a coordinate plane such that $y/x<0$ .

Concept used:

An ordered pair $\left(x,y\right)$ of real numbers to each point in a plane

A rectangular or Cartesian coordinate system in a plane by mean of two perpendicular coordinates lines called coordinate axis

Calculation:

A real number (coordinate) to each point on a line

An ordered pair $\left(x,y\right)$ of real numbers to each point in a plane

A rectangular or Cartesian coordinate system in a plane by mean of two perpendicular coordinates lines called coordinate axis that intersect at the origin O

The plane is then a coordinate plane, or an $xy$ plane such that $y/x<0$

The coordinate axes divide the plane in to four parts called first, second, third, fourth quadrants

Points on the axes do not belong to any quadrant

So,

The point $\left(x,y\right)$ in a coordinate plane such that $y/x<0$ Belong to the second or fourth quadrants.