Problem 194 - Calculus Volume 1

Question

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function. $$ f(x)=\frac{1}{x+2} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{1}{x} - 2 \). Its domain is all real numbers except 0, and its range is all real numbers except -2.

Step 1: Set Up the Equation for Inversion

To find the inverse of the function, start by replacing the function notation with \( y \). Thus, the function is \( y = \frac{1}{x+2} \). Our goal is to solve for \( x \) in terms of \( y \).

Step 2: Solve for x

Rearrange the equation to express \( x \) in terms of \( y \). Multiply both sides by \( x+2 \) to get \( y(x+2) = 1 \). Then, divide by \( y \) to isolate \( x+2 \): \( x+2 = \frac{1}{y} \). Finally, solve for \( x \): \( x = \frac{1}{y} - 2 \).

Step 3: Write the Inverse Function

Swap \( x \) and \( y \) to write the inverse function notation. The inverse function \( f^{-1}(x) \) is \( f^{-1}(x) = \frac{1}{x} - 2 \).

Step 4: Determine the Domain of the Inverse

The original function \( f(x) = \frac{1}{x+2} \) has a restriction that \( x eq -2 \) because it makes the denominator zero. For the inverse function, \( x eq 0 \) as it was in the denominator in Step 2. Therefore, the domain of \( f^{-1}(x) \) is all real numbers except 0.

Step 5: Determine the Range of the Inverse

The range of the original function excludes 0 because \( \frac{1}{x+2} = 0 \) is impossible. Therefore, the range of the inverse \( f^{-1}(x) \) is the set of all real numbers except -2.

Key Concepts

Understanding Domain and Range

The domain and range are essential aspects of any mathematical function. Simply put, the domain is the set of all possible input values (usually represented by 'x') that a function can accept. Meanwhile, the range is the set of all possible output values (usually represented by 'y') that the function can produce.

• Domain: Think of it as the "input" or "starting" values that go into the function. For the function \( f(x) = \frac{1}{x+2} \), the domain excludes \( x = -2 \) because it would result in division by zero, which is undefined.
• Range: This refers to all possible "output" values. In our exercise, the range of the original function omits 0, as there is no 'x' value that would result in a zero output in this function context.

Understanding these concepts helps in effectively analyzing and graphing functions.

Grasping Function Inversion

Function inversion is the process of finding a new function that "undoes" the effect of the original function. An inverse function, denoted as \( f^{-1}(x) \), essentially reverses the roles of inputs and outputs of the original function.

• If \( f(x) = y \), then \( f^{-1}(y) = x \).
• To find the inverse, replace \( f(x) \) with 'y' and solve the equation for 'x'.
• Swap your variables 'x' and 'y' to express the inverse function.

For instance, given \( f(x) = \frac{1}{x+2} \), solving for 'x' results in the inverse \( f^{-1}(x) = \frac{1}{x} - 2 \). This inversion switches the roles of inputs and outputs perfectly. Remembering these steps ensures accuracy when working with inverses.

A Close Look at Rational Functions

Rational functions are fractions with polynomials in the numerator and the denominator. They play a crucial role in many algebra applications.For the function \( f(x) = \frac{1}{x+2} \), it is an example of a rational function where the denominator \( x+2 \) is essential in defining the domain. Such functions have specific characteristics:

• A rational function is undefined where its denominator is zero. For our function, this occurs at \( x = -2 \).
• They often have horizontal and vertical asymptotes. For instance, the vertical asymptote here exists at \( x = -2 \), where the function value approaches infinity.

Understanding these properties enables us to grasp the behavior and graph shape of rational functions better.