Problem 116 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed $$f(x)=\left\\{\begin{array}{lll}2 & \text { if } & x \neq 4 \\\3 & \text { if } & x=4\end{array}\right.$$ and one piece of my graph is a single point.

Step-by-Step Solution

Answer

Yes, the statement makes sense. The piece of the graph which is a single point will be \( (4, 3) \), which is separated from the rest of the graph.

Step 1: Understand the Function

Notice that the function, \( f(x) = \begin{cases} 2 & \text{if } x \neq 4, \\ 3 & \text{if } x = 4 \end{cases} \), gives a specific value of 3 at \( x = 4 \) and it is equal to 2 for all other \( x \) values.

Step 2: Visualize the Graph

The graph would be a horizontal line \( y = 2 \), except for at \( x = 4 \), where there will be a point at \( y = 3 \). At \( x = 4 \), there will be two points; one lying on the line \( y = 2 \) (which is 'unreachable' because of the condition \( x \neq 4 \)) and another at \( y = 3 \). The latter creates the discontinuity point.

Step 3: Interpret the Statement

The statement indicates that one piece of the graph is a single point. This is sensible, as this refers to the defined value when \( x = 4 \), which makes it a single, separated point on the graph due to the discontinuity at \( x = 4 \).