Problem 122 - A Graphical Approach to College Algebra

Question

Restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. $$f(x)=-\sqrt{x^{2}-16}$$

Step-by-Step Solution

Answer

Restrict the domain to \([4, \infty)\) to make the function one-to-one without changing its range.

Step 1: Understanding the Function

The function given is \( f(x) = -\sqrt{x^2 - 16} \). This function involves a square root and we need to ensure the expression inside the square root, \( x^2 - 16 \), is non-negative. This is because the square root of a negative number is not defined in the real numbers.

Step 2: Determining the Domain

To keep the expression inside the square root non-negative, solve \( x^2 - 16 \geq 0 \). This means \( x^2 \geq 16 \). Therefore, \( x \leq -4 \) or \( x \geq 4 \). This gives two possible intervals for the domain: \((-\infty, -4] \cup [4, \infty)\).

Step 3: Restricting the Domain to Make the Function One-to-One

A function is one-to-one if it passes the horizontal line test, meaning no horizontal line intersects the graph at more than one point. For the given function \( f(x) = -\sqrt{x^2 - 16} \), we can consider either \( x \leq -4 \) or \( x \geq 4 \) separately as a restricted domain, ensuring the function is one-to-one.

Step 4: Choosing the Domain

We choose \([4, \infty)\) as the domain. On this interval, \( f(x) = -\sqrt{x^2 - 16} \) is one-to-one and decreasing (it passes the horizontal line test), showing that no horizontal line will intersect the graph more than once.

Step 5: Verification with a Graph

Graph \( f(x) = -\sqrt{x^2 - 16} \) for \( x \geq 4 \). Notice that the graph is a decreasing half of a parabola opening downwards, ensuring one-to-one behavior on this interval, and thus the range remains the same, \([-\infty, 0]\).

Key Concepts

One-to-One Function

A one-to-one function is a type of function where each element in the domain corresponds to exactly one unique element in the range, and vice versa. This means that no two different input values within the function's domain will map to the same output value.

To determine if a function is one-to-one, we can use the horizontal line test. If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one. Applied correctly, this test ensures that every value in the function's range relates to only one input.

For instance, the function given in the exercise, when restricted to a suitable domain, becomes one-to-one. This allows us to create a more consistent and predictable mapping between domains and ranges, important for many mathematical analyses and applications.

Square Root Function

Square root functions involve calculations with the square root of a number or expression. The defining feature of these functions is the presence of the square root symbol, typically written as \( \sqrt{...} \). It introduces a domain restriction because the expression under the square root must be non-negative for real-valued functions.

In this exercise, the function is \( f(x) = -\sqrt{x^2 - 16} \). This means \( x^2 - 16 \) must be greater than or equal to zero, leading to the requirement \( x^2 \geq 16 \). Solving this inequality gives us the domain restriction of \( x \leq -4 \) or \( x \geq 4 \). Only then is the square root of \( x^2 - 16 \) defined in real numbers.

Understanding and determining domains for square root functions is crucial since it prevents errors in calculations and graphs that involve these functions.

Horizontal Line Test

The horizontal line test is a graphical method used to determine whether a function is one-to-one. By projecting a horizontal line across the graph of the function, we check if it touches more than one point. If it does, the function does not satisfy the condition of being one-to-one.

In our exercise, we can restrict the domain to ensure the function \( f(x) = -\sqrt{x^2 - 16} \) satisfies this test. By choosing a restricted domain such as \( [4, \infty) \), the graph of the function becomes one-to-one, meaning any horizontal line will not intersect the graph more than once.

This test is especially handy when deciding on domain restrictions; it ensures that the function’s range is consistently and uniquely mapped to the domain values.

Function Graphing

Graphing functions helps visualize their behavior and understand their properties like one-to-one status and range. When graphing, it’s crucial to consider the domain restrictions that might influence the appearance and validity of the graph.

In this particular exercise, the graph of \( f(x) = -\sqrt{x^2 - 16} \) for the domain \( x \geq 4 \) shows a downward-sloping curve. Due to its parabolic nature, this section of the function graph shows it as decreasing and clearly satisfies the requirements for being a one-to-one function.

Thus, graphing supports our findings about domain restrictions and function behavior. Additionally, by graphing, we can verify that our choice of domain does not alter the expected range of \([-\infty, 0]\), ensuring that the function complies with its initial conditions and requirements.