F(x)=x+2 g(x)=x^2+4 Whats the function and relation?

1 Answer
Jul 30, 2018

#" "#
Please read the explanation.

Explanation:

#" "#
Definition of a Relation:

A Relation can be defined as a set of elements - both input and output values - available as ordered pairs.

There are no special rules.

Definition of a Function:

A Function can be defined as a set of ordered pairs in which each element of #color(red)(x# has one and only one element of #color(red)(y# associated with it.

However, it is interesting to note that two elements of #color(red)(x# may have same element of #color(red)(y# associated with them.

We are given two relations:

#color(red)(f(x)=x+2#

#color(red)(g(x)=x^2+4#

Please note that the first relation

#color(red)(f(x)=x+2#

is linear

while

the second relation

#color(red)(g(x)=x^2+4#

is quadratic

#color(blue)("What do we need to do ?"#

We need to determine whether any of these relations is a function.

We can set up the relations as tables of ordered pairs:

enter image source here

The relation #color(red)(y=f(x)=x+2# has a unique #color(red)(y# value for the corresponding element of #color(red)(x# for the ordered pairs.

Hence, this relation represents a function.

The relation

#color(red)(y=g(x)=x^2+4#

has the same #color(red)(y# value for two distinct elements of #color(red)(x#.

Hence, ordered pairs #color(red)((x,y)# are not one-to-one relationships.

It is still a function.

We can easily observe these points using their graphs:

enter image source here

There is an alternative method as well:

We can perform the vertical line test for the graphs:

enter image source here

Vertical Lines intersect the graphs exactly at one point.

Hence, the relations pass the vertical line test and we conclude that both the given relations are function.