Problem 113 - College Algebra Essentials

Question

What is the graph of a function?

Step-by-Step Solution

Answer

The graph of a function is a visualization of the relationship between input and output variables of the function on a coordinate plane. It's the set of all points for which the x-coordinates are the inputs of the function and the y-coordinates are the outputs of the function. This helps us understand and visualize the behavior of the function.

Step 1: Understanding Function

A function is a special kind of relation in which each input is related to exactly one output. In terms of Mathematics, it's a relation between a set of inputs and a set of permissible outputs, whereby each input is related to exactly one output. For example, a function \(f(x) = x^2 + 3x + 2\). The 'x' term can represent different input values, and for each of those, the function will have a different output value.

Step 2: Understanding Graph

A graph in mathematics is a tool that visualizes various types of relations between numbers. The graph is drawn on a grid called a coordinate plane. The horizontal line is called the x-axis and the vertical line is the y-axis. Every point on the graph can be represented by an ordered pair of numbers (x, y), called coordinates.

Step 3: Connecting Function and Graph

The graph of a function is the set of all points in the x-y plane for which the x-coordinates are the inputs of the function and the y-coordinates are the outputs of the function. For instance, in the function \(f(x) = x^2 + 3x + 2\), you can substitute any 'x' values to get the corresponding 'y' values, and hence, plot the graph of the function. This helps in visualizing the behavior of the function.

Step 4: Practical illustration

To better illustrate, let's take the function \(f(x) = x^2\). When we put the value of \(x = -2\), the output \(y = 4\). Similarly, when \(x = -1\), the output \(y = 1\). This way we can derive more points, plot them on the graph and then draw our graph for the function \(x^2\).