Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Work each problem. If the point \((7,8)\) lies on the graph of \(f,\) then \(F\) (_____) = (____).

Step-by-Step Solution

Answer

F(7) = 8.

Step 1: Identify the Given Point

We are given that the point \((7,8)\) lies on the graph of the function \(f\). This means when the input \(x\) is \(7\), the output \(f(x)\) is \(8\).

Step 2: Determine Function Notation

In function notation, \(f(x)\) represents the output of the function \(f\) for a given input \(x\). Here, since the point \((7,8)\) is on the graph, it tells us that \(f(7) = 8\).

Step 3: Identify the Values

Place the given point values into the blanks: The function \(F\) of \(x\) when \(x = 7\) equals \(8\). Therefore, \(F(7) = 8\).

Key Concepts

Graph of a Function

A graph of a function visually represents how each input value is transformed into an output value. Imagine a graph where the x-axis represents the input of a function and the y-axis represents the output. Each point on this graph corresponds to a pair of input and output for the function.

For example, in the function graph where the point \((7, 8)\) is plotted, this shows directly how the input value (7) is paired with the output value (8). The graph provides a clear visual way of understanding a function's behavior by showcasing this relationship.

By looking at the graph, you can easily tell what the output of a function is for any given input simply by locating its coordinates.

Coordinate Points

Coordinate points, represented as \((x, y)\), are fundamental in plotting and understanding graphs of functions. The first number in the pair, the x-coordinate, is the input. The second number, the y-coordinate, is the output.

In the given exercise, we have the point \((7, 8)\). This means when the input equals 7, the output of the function is 8. These coordinate points are crucial for graphing functions, as they map out precisely where points lie on a plane.

Such a coordinate point ensures that each specific input will have one specific output, which helps in establishing whether a graph represents a function.

Input and Output

In mathematics, a function describes a specific relationship where each input has a single output. This relationship is often written in function notation where \(f(x)\) denotes the function's output for a given input \(x\).

Referring back to our example, the point \((7, 8)\) tells us that if our input \(x\) is 7, then our function's output, or \(f(7)\), is 8. From this notation, we can easily determine what the function does to the input to produce the output.

Understanding input and output is critical as it forms the foundation of how functions operate, allowing us to predictably and consistently translate inputs into outputs with the function formula.