Problem 16 - Calculus of a Single Variable

Question

Use a graphing utility to graph the function. Then use the Horizontal Line Test to determine whether the function is one-to-one on its entire domain and therefore has an inverse function. \(f(x)=\frac{6 x}{x^{2}+4}\)

Step-by-Step Solution

Answer

The function \(f(x)=\frac{6x}{x^{2}+4}\) is one-to-one over its entire domain and thus, it has an inverse function.

Step 1: Understanding the Function

Look at the function \(f(x)=\frac{6x}{x^{2}+4}\). This is a rational function where the numerator is a first-degree polynomial and the denominator is a second-degree polynomial. The domain of this function is all real numbers, since the denominator can never be zero.

Step 2: Graphing the Function

Use a graphing utility to sketch the function. It is seen that the function appears to be decreasing across its entire domain, and no horizontal line intersects the graph of the function more than once.

Step 3: Applying the Horizontal Line Test

Check using the Horizontal Line Test, if every horizontal line intersects the graph in at most one point, the function is called one-to-one. Since the function graph does indeed pass the horizontal line test, the function is one-to-one and thus has an inverse.

Key Concepts

Horizontal Line Test

The Horizontal Line Test is a technique to determine if a function is one-to-one. When you draw a horizontal line across a graph, check if any line intersects with the curve at more than one point. This is the key.
If a horizontal line touches the curve more than once, the function is not one-to-one. If the horizontal line only hits the curve once, then the function is one-to-one.

• One-to-one functions allow for an inverse function.
• It's crucial to check the entire domain of the function.
• A graph that passes this test has a unique output for every input.

In simple words, the horizontal line test helps us identify whether each output value is achieved by only one input value. This quality is highly important in ensuring that inverses are possible for these functions.

One-to-One Function

A one-to-one function, often referred to as injective, maps every element of its domain to a unique element in its co-domain. This means no two different inputs give the same output.
Mathematically, a function \(f\) is one-to-one if \(f(a) = f(b)\) implies \(a = b\). In simpler terms, every 'x' has its unique 'y'.

• These functions have inverses because every input has a distinct output.
• Graphically identified using the Horizontal Line Test.
• Ensures that invertibility is preserved across its domain.

Recognizing a one-to-one function is essential for understanding inverse functions, as it provides a solid foundation for reversing calculations and functions.

Rational Functions

Rational functions are ratios of two polynomials. The function \(f(x) = \frac{6x}{x^2+4}\) is a classic example. They often include variations that look like fractions.
In these functions, the denominator cannot be zero, which sometimes affects the domain, although not in this specific case because \(x^2 + 4\) is never zero.

• Key feature: Complex behavior over different domains.
• Has vertical and horizontal asymptotes that characterize its limits.
• Often goes into infinity or negative infinity at vertical asymptotes.

Rational functions can display unique behavior, such as curves, bends, and approaches, which govern how they appear and are studied within mathematical contexts. Understanding these intricacies allows for a deeper grasp of their graphical and functional properties.