Chapter 1.7, Problem 51PPE

To determine

To find: If the given relation $y=\frac{x}{2}-6$ is a function or not.

Answer to Problem 51PPE

Yes, given relation is a function because for two different input values, there is always a different unique output value. And given one is a linear function actually as its degree is one.

Explanation of Solution

Given information:

Given a relation, $y=\frac{x}{2}-6$ .

Concept used:

A relation is a function,

• If for each value in its domain; there is always a unique value in its co domain.
• If for each different first value of its ordered pair, there is always a second unique value, then such relation is a function always.
• Or, no two same first values, should have two different second values in ordered pairs of set of relation.

Calculation:

In tabular form, given relation can be written in its ordered pair form as

x	$y=\frac{x}{2}-6$	$(x,y)$
0	$y=\frac{0}{2}-6$	$(0,-6)$
2	$y=\frac{2}{2}-6=1-6=-5$	$(2,-5)$
4	$y=\frac{4}{2}-6=2-6=-4$	$(4,-4)$
12	$y=\frac{12}{2}-6=6-6=0$	$(12,0)$

Or, given relation as a set of ordered pairs can be represented as :

  $R=\{\left(0,-6\right),\left(2,-5\right),\left(4,-4\right)\dots .\}$

So, for each different input value of x, there is always a unique output value of variable y. That satisfies the definition of a function.

Conclusion:

In above relation, for each different first value, there is always a unique second value in its given ordered pairs. So, it is surely a function, based on definition of the function.