Problem 71 - A Graphical Approach to College Algebra

Question

Work each problem. If \(f(-2)=3,\) identify a point on the graph of \(f\)

Step-by-Step Solution

Answer

The point is \((-2, 3)\).

Step 1: Understand the Given Information

We are given that when the input to the function \( f \) is \(-2\), the output is \(3\). This is written as \( f(-2) = 3 \).

Step 2: Identify the Point on the Graph

A point on the graph of a function \( f \) is represented as an ordered pair \((x, f(x))\). In this case, the input \(x\) is \(-2\) and the output \(f(x)\) is \(3\).

Step 3: Formulate the Ordered Pair

Using the information from the previous steps, the point on the graph is \((-2, 3)\).

Key Concepts

Ordered Pairs

When visualizing mathematical concepts on a graph, we often use ordered pairs. An ordered pair is simply a way to represent two quantities as a set of numbers within parentheses, such as \( (x, y) \). In the context of graphing functions:

• The first number, \( x \), is the input.
• The second number, \( y \), is the output, usually represented as \( f(x) \) or another function name.

Ordered pairs follow a specific order where the first position is always associated with the x-coordinate or input, and the second with the y-coordinate or output. For example, in our exercise, the ordered pair \( (-2, 3) \) informs us that when \( -2 \) is plugged into function \( f \), we get an output of \( 3 \). This point is crucial in graphing as it shows a specific location on the plane that the function passes through.
Understanding ordered pairs helps in constructing and interpreting graphs, functioning as the building blocks of plotting any function accurately.

Input and Output

The relationship between inputs and outputs is fundamental in the study of functions. In a function, each input corresponds to exactly one output. Here's how to understand this concept:

• **Input**: The value you provide to a function. In our example, the input is \( -2 \).
• **Output**: What you get after applying the function to the input. Here, the output for \( -2 \) is \( 3 \).

This input-output relationship is often expressed via notation, such as \( f(x) \), where \( f \) is the function's name and \( x \) is the variable or input. The output is then the function evaluated at that input value, like \( f(-2) = 3 \).
Understanding inputs and outputs allows us to predict behavior and understand the flow of mathematical calculations in various scenarios. This concept is very powerful, as it translates real-world scenarios into mathematical models, making complex systems easier to manage and visualize.

Graphing Functions

Graphing functions involves representing the relationship between inputs and outputs visually on a coordinate grid. Here's a simple way to break it down:

• The x-axis (horizontal) represents the input values.
• The y-axis (vertical) shows the corresponding output values.
• A function graph is made up of points that satisfy the function's equation, given by ordered pairs \( (x, f(x)) \).

To graph a function, you plot each ordered pair and then connect the points smoothly if they form a continuous line or curve. For example, if you have a point \( (-2, 3) \), you place a point that aligns with \( x = -2 \) on the x-axis and \( y = 3 \) on the y-axis.
This visual representation aids in understanding how the function behaves across different input values, identifying patterns, and solving problems through interpretation of shapes and slopes. Graphing is an essential technique in mathematics, used extensively in various fields such as physics, economics, and engineering.