Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each function is one-to-one. $$y=\frac{-4}{x-8}$$

Step-by-Step Solution

Answer

The function is one-to-one.

Step 1: Understand One-to-One Functions

A function is one-to-one if each element of the range is mapped to by exactly one element of the domain. This means that no two different x-values can produce the same y-value in one-to-one functions.

Step 2: Investigate the Function

Consider the function given: \( y = \frac{-4}{x-8} \). This is a rational function. To determine if it is one-to-one, we need to check if different x-values lead to different y-values.

Step 3: Horizontal Line Test

For a function to be one-to-one, no horizontal line should intersect the graph of the function at more than one point. For the given rational function, \( y = \frac{-4}{x-8} \), evaluate by imagining any horizontal line \( y = k \). Each \( y = k \) intersects the graph at only one x-value, since the function can't output the same y-value for different x-values.

Step 4: Algebraic Test for Injectivity

Test algebraically by assuming \( y_1 = y_2 \), which results from different x-values \( x_1 \) and \( x_2 \). Set \( \frac{-4}{x_1 - 8} = \frac{-4}{x_2 - 8} \). This implies \( x_1 - 8 = x_2 - 8 \), hence \( x_1 = x_2 \). Therefore, the function is injective.

Key Concepts

Horizontal Line Test

The Horizontal Line Test is a simple and intuitive way to determine if a function is one-to-one, also known as injective. Imagine drawing horizontal lines across the graph of the function. A function is considered one-to-one if and only if no horizontal line intersects the graph more than once.

For the function given in the exercise, \( y = \frac{-4}{x-8} \), applying the Horizontal Line Test requires envisioning any possible horizontal line at \( y = k \), where \( k \) is any constant. In this rational function, any horizontal line will intersect the curve only once as each \( y \) value results from a unique \( x \) value.

In simpler terms:

• If you can draw a horizontal line that hits the function’s graph in more than one spot, the function isn't one-to-one.
• If every horizontal line you can draw only hits one point, the function is one-to-one.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. In the expression \( y = \frac{-4}{x-8} \), \(-4\) is the numerator, and \( x-8 \) is the denominator.

These kinds of functions are important as they exhibit interesting behaviors, like vertical asymptotes, and can model several real-world phenomena. Let's explore their characteristics:

• Rational functions are undefined where the denominator equals zero. For \( y = \frac{-4}{x-8} \), the function is undefined at \( x = 8 \) because dividing by zero is not allowed.
• They might have vertical asymptotes, as is the case here at \( x = 8 \).
• The graph of a rational function can display unique patterns making them easy to distinguish from linear or polynomial functions.
• Rational functions are typically smooth, except at points where they are undefined, resulting in vertical asymptotes.

Injectivity

Injectivity, or being one-to-one, is a property of a function indicating that distinct inputs (x-values) lead to distinct outputs (y-values). This is a crucial trait for defining the inverse of a function.

In mathematical language, a function \( f : A \rightarrow B \) is injective if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \). In the exercise, to show \( y = \frac{-4}{x-8} \) is injective, we set \( \frac{-4}{x_1-8} = \frac{-4}{x_2-8} \). By algebraic manipulation, this tells us \( x_1 = x_2 \), confirming injectivity.

Here's an easy way to remember it:

• No two different inputs should result in the same output.
• If you test several inputs and see that no output repeats, the function is injective.

Understanding injectivity helps in grasping the core concepts of function behavior and how they map elements from one set to another.