Problem 2 - Precalculus Mathematics for Calculus

Question

If \(f(2)=3,\) then the point \((2, _______ )\) is on the graph of \(f\).

Step-by-Step Solution

Answer

(2, 3)

Step 1: Identify the Known Variable

The problem states that when the input to the function is 2, the output is 3. This is written as \( f(2) = 3 \).

Step 2: Determine the Y-Coordinate

For a point \((x, y)\) to lie on the graph of a function \(f\), the y-coordinate is the output of the function when the input is \(x\). In this case, the input \(x\) is 2, and the function output \(y\) from the given \(f(2) = 3\) is 3.

Step 3: Fill in the Missing Coordinate

Since \(f(2) = 3\), the y-coordinate of the point on the graph is 3. Therefore, the complete point is \((2, 3)\).

Key Concepts

Coordinates

In mathematics, coordinates are the numbers that define the location of a point on a graph. They are crucial for graphing functions and understanding spatial relations in geometry. Typically, coordinates are expressed as pairs

• For a two-dimensional plane, coordinates take the form \((x, y)\).
• The first number, \(x\), represents the horizontal position (often referred to as the "input" in the context of functions).
• The second number, \(y\), is the vertical position (or the "output" in a function graph).

Understanding these coordinates helps you locate points accurately on a graph. In the case of the function \(f(x)\), if given \(f(2) = 3\), this implies that the point \((2, 3)\) is situated on the function's graph, where 2 is the \(x\)-coordinate and 3 is the \(y\)-coordinate.

Function Output

The output of a function, often represented by \(y\) in the coordinates \((x, y)\), is the result we obtain when we substitute a specific value into the function. It is crucial to understand this concept when dealing with function graphs.

• Consider the function \(f(x)\), which acts like a machine that takes an input \(x\) and transforms it into an output \(f(x)\), or \(y\).
• In our exercise, with \(f(2) = 3\), when the input \(x = 2\), the output is \(3\).

This output signifies the point's vertical position on the graph, which, when plotted, acts as the \(y\)-coordinate. Thus, knowing the function's output allows you to plot each point precisely on its graph.

Points on Graph

Mapping points on a graph is like plotting stars on a map of the sky. Each point on the graph of a function represents a specific pair of coordinates \((x, y)\). To correctly locate a point, follow these guidelines:

• Identify your coordinates \((x, y)\), where \(x\) is the input and \(y\) is the function's output.
• Using \(x\) and \(y\), you can accurately plot the point on the graph.

In the given scenario, since \(f(2) = 3\), the specific point \((2, 3)\) lies on the graph of the function \(f\). This means if you were to draw the graph on a plane, you would plot the point by moving two units along the horizontal axis and three units up the vertical axis. Understanding how to identify and plot these points is fundamental for visualizing and analyzing function graphs.