Problem 72 - A Graphical Approach to College Algebra

Question

Work each problem. If \(f(3)=-9.7,\) identify a point on the graph of \(f .\)

Step-by-Step Solution

Answer

The point on the graph is (3, -9.7).

Step 1: Understand the Function Evaluation

Given that \( f(3) = -9.7 \), it indicates that when \( x = 3 \), the function \( f \) outputs the value \( -9.7 \).

Step 2: Identify the Coordinates

For any function \( f \), if \( f(x) = y \), then \( (x, y) \) is a point on its graph. With \( f(3) = -9.7 \), the coordinates of this point are \( (3, -9.7) \).

Key Concepts

Graph of a Function

A graph of a function is a visual representation of a relationship defined between two variables, usually represented as the x-axis and y-axis in a coordinate plane. When we say a graph of a function, it connects all the points that satisfy the function equation. This connection forms a line or curve depending on the function's behavior.
For example, when plotting the graph of a function like the one given in our problem, you map out each point that makes the function true. If the function is given as \( f(x) \), then every point \( (x, f(x)) \) on the graph is a solution.

• A straight line indicates a linear function.
• A curve might indicate a quadratic or higher-degree function.

Visualizing functions through graphs is a powerful tool for understanding the nature of equations and their solutions. It allows us to see the trends, such as whether a function is increasing or decreasing, constant, or oscillating.

Coordinates

Coordinates on a graph represent specific points with a particular value for each variable. In a 2D plane, points are determined using an ordered pair \((x, y)\), where \(x\) is the horizontal position, and \(y\) is the vertical position.
For example, if we say a point is \((3, -9.7)\), the coordinate \(3\) is along the x-axis, and the coordinate \(-9.7\) is along the y-axis. This tells us precisely where the point is located. Every point on the graph of a function can be identified in this way.
Coordinates bridge the actual value of the function for a specific \(x\) to its graphical representation. They make the function not only abstractly understand but visually interpretable.

Function Notation

Function notation is a compact way of indicating a function and the operations applied to its inputs. Typically written as \(f(x)\), it describes a function named \(f\) that assigns to each \(x\) an output \(f(x)\).
This notation is essential because it clarifies which variable is being manipulated and what output should be expected.

• In our example, \(f(3) = -9.7\) indicates that input \(3\) gives the function's output \(-9.7\).
• This highlights the function's role as a machine that takes an input and produces a precise output.

Function notation simplifies discussions on functions and is used to evaluate and pinpoint specific values across various domains mathematically. It also aids in composing and manipulating multiple functions systematically.