Problem 1 - College Algebra with Modeling and Visualization

Question

Evaluating and Representing Functions If \(f(-2)=3,\) identify a point on the graph of \(f.\)

Step-by-Step Solution

Answer

The point on the graph is \((-2, 3)\).

Step 1: Understand Function Notation

The function notation \(f(-2) = 3\) means that when \(x=-2\), the output or value of the function \(f(x)\) is 3. In simpler terms, plugging \(-2\) into the function gives us \(3\) as a result.

Step 2: Identify the Point on the Graph

A point on the graph of a function \(f\) is typically represented as \((x, f(x))\). From the information given, \(f(-2) = 3\), so the point on the graph is \((-2, 3)\).

Key Concepts

Function Notation

When dealing with functions, you often see expressions like \(f(x)\) where \(f\) is the name of the function, and \(x\) is the input variable. This is called function notation. It allows us to clearly express what a function does to an input to produce an output. For example, in the notation \(f(-2) = 3\), \(-2\) is the input, and \(3\) is the output. This tells us that if we input \(-2\) into function \(f\), it responds with the value \(3\).

Function notation enables us to easily plug in different values for \(x\) and find the corresponding outputs. This process is known as evaluating the function. Understanding how to read and use function notation is crucial for solving problems involving functions, as it provides a standardized way to communicate complex relationships between variables.

Graph of a Function

A graph of a function visually represents the relationship between input values (often denoted on the x-axis) and their corresponding outputs (often on the y-axis). When plotting a function on a graph, every possible input and its output pair is a unique point, creating a graphical curve or line. The graph of the function \(f(x)\) is a collection of points \((x, f(x))\). This visual representation helps us understand the behavior of functions at a glance.

For example, if you know that \(f(-2) = 3\), on the graph of the function \(f(x)\), you would mark the point \((-2, 3)\). This coordinates tell you that when \(x\) is \(-2\), the function takes on the value \(3\). By analyzing the graph, you can see trends, such as whether a function is increasing or decreasing, and detect any maximum or minimum values without calculation.

Coordinate Points

Coordinate points are the pairs \((x, y)\) that are used to plot locations on a graph. In the context of a function, a coordinate point \((x, f(x))\) tells us the output of the function for a particular input \(x\). Think of each coordinate point as a map location. It tells you exactly where the function touches the graph based on given input and output values.

For instance, from the exercise, the function evaluates to \(f(-2) = 3\), so the coordinate point \((-2, 3)\) relates the input \(-2\) with its output \(3\). Graphically speaking, when you plot this point, it gives a clear visual representation of this mathematical relationship. Coordinate points are essential for constructing graphs of functions or understanding how individual elements of a function behave individually and collectively.