Chapter 4.3, Problem 21PPE

To determine

The rule that represent the function of given ordered pairs, $\left(0,\frac{2}{19}\right),\left(1,1\frac{2}{19}\right),\left(2,4\frac{2}{19}\right),\left(3,9\frac{2}{19}\right),\left(4,16\frac{2}{19}\right)\text{and }\left(5,25\frac{2}{19}\right)$

Answer to Problem 21PPE

The rule of the function is $y=x^2+\frac{1}{19}$

Explanation of Solution

Given:

The given ordered pair is,

  $\left(0,\frac{2}{19}\right),\left(1,1\frac{2}{19}\right),\left(2,4\frac{2}{19}\right),\left(3,9\frac{2}{19}\right),\left(4,16\frac{2}{19}\right)\text{and }\left(5,25\frac{2}{19}\right)$

First represent the given ordered pair in a tabular form.

x	0	1	2	3	4	5
y	$\frac{2}{19}$	$1\frac{2}{19}$	$4\frac{2}{19}$	$9\frac{2}{19}$	$16\frac{2}{19}$	$25\frac{2}{19}$

Now, for first column

For x = 0, $y=\frac{2}{19}$

It can be represented by the rule $y=x^2+\frac{2}{19}\text{and}y=x+\frac{2}{19}$ only.

Now, checking the rule $y=x+\frac{2}{19}$ for second column. So, put x = 2

  $\begin{array}{l}y=x+\frac{2}{19}\\ =2+\frac{2}{19}\\ =2\frac{2}{19}\end{array}$

So $y=x+\frac{1}{19}$ is not same for second column.

Now, checking the rule $y=x^2+\frac{2}{19}$ for second column

Put x = 2

  $\begin{array}{l}y=x^2+\frac{2}{19}\\ =2^2+\frac{2}{19}\\ =4\frac{2}{19}\end{array}$

So, $y=x^2+\frac{2}{19}$ is same for second column.

Hence, only $y=x^2+\frac{2}{19}$ rule follows for second column.

Similarly, it is same for the remaining column also.

Hence, the rule of the function for the given ordered pair is $y=x^2+\frac{2}{19}$