Problem 60 - College Algebra with Modeling and Visualization

Question

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{1}{x+5}+2 $$

Step-by-Step Solution

Answer

\(f^{-1}(x) = \frac{1}{x-2} - 5\)

Step 1: Understand the Inverse Function

The inverse function, denoted as \(f^{-1}(x)\), reverses the process of the original function \(f(x)\). This means if \(y = f(x)\), then \(x = f^{-1}(y)\).

Step 2: Set Up the Equation

Given the function \(f(x) = \frac{1}{x+5} + 2\), start by letting \(y = f(x)\). So, \(y = \frac{1}{x+5} + 2\). Our goal is to solve this equation for \(x\) in terms of \(y\).

Step 3: Isolate the Fraction

Subtract 2 from both sides to isolate the fraction: \[ y - 2 = \frac{1}{x+5} \]

Step 4: Solve for the Reciprocal

Take the reciprocal of both sides to solve for \(x + 5\): \[ \frac{1}{y-2} = x+5 \]

Step 5: Solve for x

Subtract 5 from both sides to solve for \(x\): \[ x = \frac{1}{y-2} - 5 \]

Step 6: Write the Inverse Function

The inverse function \(f^{-1}(x)\) is obtained by letting \(y = x\): \[ f^{-1}(x) = \frac{1}{x-2} - 5 \]

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's essentially about finding the unknown or putting real-life variables into equations and then solving them. In this exercise, we're particularly interested in reversing a process using algebra: finding the inverse of a function.

An inverse function is essentially the reverse of the original function. For instance, if the function describes how to get from point A to B, the inverse function will show you how to get back to A from B. In algebra, finding the inverse function is about solving the equation of the original function for a different variable.

Here, with the function\[ f(x) = \frac{1}{x+5} + 2 \]we're looking for a way to express this relationship in reverse. To find the inverse, we set up the equation\[y = f(x) = \frac{1}{x+5} + 2 \]and solve for \(x\) in terms of \(y\). This involves several steps, including isolating fractions and manipulating algebraic expressions, which are typical operations in algebra to solve for the desired variable.

Function Operations

Function operations allow us to perform various algebraic transformations on functions. When dealing with inverse functions, we primarily perform operations to "reverse" the function. This involves swapping roles between the dependent and independent variables and following a series of algebraic steps to isolate the original input variable.

To understand this, let us consider the function given:\[ y = \frac{1}{x+5} + 2 \]Function operations guide us to

• Subtract 2 from both sides, isolating the fraction.
• Take the reciprocal of both sides, reversing the division relationship.
• Finally, adjust terms by subtraction to solve for the original input variable.

These operations are essential to finding the inverse as each step carefully undoes the operations applied in the function.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. The function in this exercise features a rational expression,\[ \frac{1}{x+5} + 2\]Understanding and manipulating rational expressions is crucial when finding the inverse function since reversing these operations is vital.

The key operations involved include:

• Isolating the rational part of the expression by subtracting constant terms.
• Manipulating the expression by performing reciprocal operations to verify input variable.

Rational expressions require careful attention - inaccuracies in operations such as subtracting constants or taking reciprocals can dramatically affect outcomes. In this exercise, correctly managing these expressions allows us to derive the inverse function formula:\[ f^{-1}(x) = \frac{1}{x-2} - 5\]This illustrates how rational expressions play a fundamental role in understanding and solving function inverses.