Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Draw the graph of a function \(f\) that is 1 -to- 1 and a function \(g\) that is not 1 -to-1.

Step-by-Step Solution

Answer

Function \(f(x) = x\) is an example of a one-to-one function. Its graph is a straight line passing through the origin, and it passes the horizontal line test. Function \(g(x) = x^2\), on the other hand, is a function that is not one-to-one. Its graph is a parabola and fails the horizontal line test.

Step 1: Draw Function f

To begin with, draw a function \(f\) which is 1 -to- 1. An example of such a function could be \(f(x) = x\). To represent this graphically, plot a straight diagonal line passing through the origin on the Cartesian plane.

Step 2: Check the One-to-One Property of Function f

One way to check if a function is one-to-one is by applying the horizontal line test. If any horizontal line passes through the graph of the function more than once, then the function is not one-to-one. For function f, as it is a straight line, any horizontal line will only touch the graph at a single point, therefore it passes the horizontal line test and is one-to-one.

Step 3: Draw Function g

Next, a function \(g\) that is not 1 -to-1 is required. An example can be \(g(x) = x^2\). To illustrate this visually, plot a parabola on the Cartesian plane.

Step 4: Check the One-to-One Property of Function g

Again, apply the horizontal line test to function \(g\). For \(g(x) = x^2\), a horizontal line will intersect the graph at more than one point. Therefore, it fails the horizontal line test, proving that function g is not one-to-one.

Key Concepts

Horizontal Line Test

The horizontal line test is a useful tool to determine if a function is one-to-one. When a function is one-to-one, each output value is linked to only one input value. To perform the test:

• Imagine drawing various horizontal lines (parallel to the x-axis) across the graph of the function.
• Observe if any line intersects the graph at more than one point.

If a horizontal line only crosses the graph at a single point each time, the function is one-to-one. However, if any horizontal line crosses the graph more than once, the function is not one-to-one. For example, the graph of the function \(f(x) = x\) will only be touched once per horizontal line, confirming it's one-to-one. In contrast, the function \(g(x) = x^2\) would be intersected twice by many horizontal lines, demonstrating it's not one-to-one.

Function Graphs

Understanding function graphs is crucial when analyzing different mathematical functions. A function graph visually represents the relationship between inputs (x-values) and outputs (y-values). By plotting points on a graph:

• We can observe patterns or shapes that help identify the type of function.
• These visualizations can provide insights into the nature of the function, such as symmetry, intercepts, and behavior at infinity.

For instance, linear functions, like \(f(x) = x\), are depicted as straight lines. Meanwhile, quadratic functions, like \(g(x) = x^2\), form parabolas that open upwards. By analyzing these graphs, we can apply tests, like the horizontal line test, to determine properties such as the function’s one-to-one nature.

Parabolas

Parabolas are distinctive U-shaped graphs often associated with quadratic functions. The standard form of a quadratic function is \(y = ax^2 + bx + c\). In the context of one-to-one functions:

• Parabolas are crucial because they often fail the horizontal line test.
• They are symmetrical, meaning that for most horizontal lines, there will be two points of intersection for the same y-value.

This symmetry showcases why parabolas like the graph of \(g(x) = x^2\) are not one-to-one. However, restricting the domain (considering only one side of the parabola) can turn such a function into a one-to-one mapping by focusing on one unique portion of the graph, though this breaks its natural form.

Linear Functions

Linear functions are one of the simplest types of functions, represented graphically as straight lines. Characterized by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept:

• They have a constant rate of change and are often easy to analyze.
• For any linear function, a horizontal line will intersect at most one point, ensuring it is always one-to-one.

An example of a one-to-one linear function is \(f(x) = x\), which passes through the origin with a slope of one. Unlike parabolas, linear graphs possess no symmetry that would result in double intersections with any horizontal line, solidifying their one-to-one status automatically.