Problem 91 - Algebra and Trigonometry

Question

Finding an Inverse "in Your Head" In the margin notes in this section we pointed out that the inverse of a function can be found by simply reversing the operations that make up the function. For instance, in Example 6 we saw that the inverse of $$ f(x)=3 x-2 \quad \text { is } \quad f^{-1}(x)=\frac{x+2}{3} $$ because the "reverse" of "Multiply by 3 and subtract \(2^{\prime \prime}\) is "Add 2 and divide by 3 ." Use the same procedure to find the inverse of the following functions. $$ \begin{array}{ll}{\text { (a) } f(x)=\frac{2 x+1}{5}} & {\text { (b) } f(x)=3-\frac{1}{x}} \\ {\text { (c) } f(x)=\sqrt{x^{3}+2}} & {\text { (d) } f(x)=(2 x-5)^{3}}\end{array} $$ Now consider another function: $$ f(x)=x^{3}+2 x+6 $$ Is it possible to use the same sort of simple reversal of operations to find the inverse of this function? If so, do it. If not, explain what is different about this function that makes this task difficult.

Step-by-Step Solution

Answer

Reversing operations for \(f(x) = x^3 + 2x + 6\) isn't straightforward due to the polynomial form. Numerical or complex methods are required.

Step 1: Understanding Reverse Operations

To find the inverse of a function, identify and reverse the computation steps used in the function. This will involve reversing the order and operations like addition, subtraction, multiplication, division, and other transformations.

Step 2: Inverse of Function (a)

The given function is \(f(x) = \frac{2x + 1}{5}\). Reverse the operations by first multiplying by 5 (to undo the division by 5) and then subtracting 1 (to undo the addition of 1):1. Multiply by 5: \(5f(x) = 2x + 1\)2. Subtract by 1: \(5f(x) - 1 = 2x\)3. Divide by 2: \(x = \frac{5f(x) - 1}{2}\)Thus, the inverse is \(f^{-1}(x) = \frac{5x - 1}{2}\).

Step 3: Inverse of Function (b)

The given function is \(f(x) = 3 - \frac{1}{x}\). Reverse the operations as follows:1. Subtract 3 to undo the addition: \(x - 3 = -\frac{1}{x}\)2. Take the reciprocal to solve for \(x\): \(-\frac{1}{x-3} = x\)Thus, the inverse is \(f^{-1}(x) = \frac{-1}{x-3}\).

Step 4: Inverse of Function (c)

The given function is \(f(x) = \sqrt{x^3 + 2}\). First, reverse the operations:1. Square both sides to undo the square root: \((f(x))^2 = x^3 + 2\)2. Subtract 2: \((f(x))^2 - 2 = x^3\)3. Take the cube root: \(x = \sqrt[3]{(f(x))^2 - 2}\)Thus, the inverse is \(f^{-1}(x) = \sqrt[3]{x^2 - 2}\).

Step 5: Inverse of Function (d)

The given function is \(f(x) = (2x - 5)^3\). Reverse the operations:1. Take the cube root: \(\sqrt[3]{f(x)} = 2x - 5\)2. Add 5: \(\sqrt[3]{f(x)} + 5 = 2x\)3. Divide by 2: \(x = \frac{\sqrt[3]{f(x)} + 5}{2}\)Thus, the inverse is \(f^{-1}(x) = \frac{\sqrt[3]{x} + 5}{2}\).

Step 6: Analyzing Function

The function \(f(x) = x^3 + 2x + 6\) is given. To find its inverse, we'd need to reverse its operations. However, the combination of \(x^3\), \(2x\), and a constant 6 makes it a non-trivial polynomial equation that can't be easily reversed using simple algebraic manipulations. Solving for \(x\) in terms of \(f(x)\) is not straightforward due to the presence of both quadratic and linear terms. This typically requires numerical methods or complex algebraic solutions, explaining the challenge in finding a simple reversal.

Key Concepts

Function Operations

Function operations refer to actions like addition, subtraction, multiplication, and division performed on functions. These operations are essential in manipulating and simplifying mathematical expressions. They help in finding inverses or combining functions. When dealing with function operations, it’s important to keep the sequence of operations clear. Remember the order of operations (PEMDAS - Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This is crucial for successfully managing more complex functions.
For example, if you have a function, say, \(f(x) = \frac{2x + 1}{5}\), you perform division before addition. To reverse this, the first step is to multiply by 5, then subtract 1. Understanding these well-ordered steps simplifies the process to find inverses and handle polynomial functions with ease.

Inverse Operations

To find the inverse of a function, you need to reverse the operations used to compute the original function. This process involves interchanging the roles of the function and its input variable. In other words, if your original function is \(f(x) = y\), you need to rearrange everything to express \(x\) in terms of \(y\).
For instance, consider the function \(f(x) = 3 - \frac{1}{x}\). To find the inverse, think about doing the opposite of each operation: subtract 3 becomes "+3", and taking the reciprocal switches places. Thus, the inverse can be expressed as \(f^{-1}(x) = \frac{-1}{x-3}\). This method is excellent for simpler functions but can be more complicated with polynomial functions that have more terms and operations involved.

Polynomial Functions

Polynomial functions are expressions constructed from variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. They are among the most important concepts as they can describe a wide range of phenomena in science and engineering. Common forms include quadratic and cubic equations.
These functions, like \(f(x) = x^3 + 2x + 6\), present challenges for finding their inverses algebraically due to their complexity. Solving these involves understanding their structure and often applying techniques like factoring or using the Rational Root Theorem. However, some polynomials don't have a simple inverse, highlighting why operations with these functions require an advanced approach.

Solving Equations

Solving equations is all about finding the value of variables that make the equation true. This often involves isolating the variable by performing inverse operations. The goal is to simplify the equation until you can clearly see the solution.
A key aspect is being able to rearrange and manipulate equations involving different types of functions. For example, in solving polynomial equations like \(f(x) = x^3 + 2x + 6\), there is no straightforward algebraic way to find the inverse due to combining powers and terms. Instead, solving might necessitate iterative methods or numerical approaches. This complexity is often not covered deeply in basic studies but grows in importance with more advanced mathematics. Understanding these concepts forms the groundwork for tackling more challenging mathematical problems.