Question

Determine whether each relation is a function. Explain.

$y=\frac{1}{2}x-6$

Short answer

For two different input values, there is always a different unique output value. Thus the given relation is a function. This is a linear function as its degree is one.

Step-by-step solution

Step 1. Given.

The given relation is, $y=\frac{x}{2}-6$.

Step 2. Concept.

A relation is called a function if the following conditions are satisfied:

1. Every value in its domain has a unique value in its codomain.
2. For every value of x, there is a unique value of y.
3. For every value of x, there must be only one value of y.

Step 3. Calculation.

The given relation can be written in its ordered pair form as following

$x$	$y=\frac{x}{2}-6$	$(x,y)$
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, the given relation as a set of ordered pairs becomes:

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

Therefore, for each different value of x, there is always a unique value of y. That satisfies the definition of a function. Thus, the given relation is a function.