Q 40. - Precalculus Enhanced with Graphing Utilities

Question

Is a graph that consists of a single point the graph of a function? Can you write the equation of such a function?

Step-by-Step Solution

Answer

Yes, a graph that consists of a single point is the graph of a function.

For example: f(x) =  3 is a function if x = 2 otherwise it is undefined.

Step 1. Definition used.

A relation from a set of inputs to a set of possible outputs where each input is related to exactly one output is called as a function.

Step 2. Concept used.

Vertical line test:

If all vertical lines intersect a curve at most once then the curve represents a function.

Step 3. Explanation.

Yes, 

A single point on the graph is a function.

We know that every input in the domain has exactly one output.

Suppose $(m , n)$ is the point on the graph.

Graphically if vertical line cross the graph at most  once then the graph is the graph of a function.

If we consider the point $(m , n)$and we can draw the vertical line y = n then the point $(m , n)$ is represent a function otherwise it is undefined.

For example: Let (2, 3) is a point on the graph. 

and f(x) =  3 is a function if x = 2 otherwise it is undefined.

Step 4. Conclusion.

Yes, a graph that consists of a single point is the graph of a function.

For example: f(x) =  3 is a function if x = 2 otherwise it is undefined.