Problem 20 - Contemporary Precalculus

Question

Use algebra to find the inverse of the given one-to-one function. $$f(x)=\frac{x}{x+1}$$

Step-by-Step Solution

Answer

Answer: The inverse of the given one-to-one function is f^(-1)(x) = (-x) / (x - 1).

Step 1: Replace the function notation with a variable

Replace f(x) with the variable y. $$y=\frac{x}{x+1}$$

Step 2: Swap the roles of x and y

Switch the places of x and y in the equation: $$x=\frac{y}{y+1}$$

Step 3: Solve for y

We want to isolate y, so first multiply both sides by (y+1) to eliminate the denominator: $$(y+1)x=y$$ Now expand the left side of the equation and move all the terms containing y to one side to make it easier to isolate y: $$xy+x-y=0$$ Rearrange the equation to make it clearer: $$y(x-1)=-x$$ Now divide both sides by (x-1) to solve for y: $$y=\frac{-x}{x-1}$$

Step 4: Replace y with the inverse function notation

Replace y with the inverse function notation, f^(-1)(x): $$f^{-1}(x)=\frac{-x}{x-1}$$ So, the inverse of the given one-to-one function is: $$f^{-1}(x)=\frac{-x}{x-1}$$

Key Concepts

One-to-One Functions

A one-to-one function is a special kind of function where each output value is matched with one unique input value. This means if you have a function value, you can trace it back to exactly one input. This property is essential when finding inverse functions because it ensures that the inverse will also be a valid function.
To identify a one-to-one function, you can use the Horizontal Line Test. If any horizontal line cuts the graph of your function at most once, then the function is one-to-one.
Understanding one-to-one functions helps you confidently find their inverses since the process will yield a unique solution that is a function itself.

Algebraic Manipulation

Algebraic manipulation involves changing the form of an equation or expression to simplify it or solve for a variable of interest. It is a crucial skill when finding inverse functions, as it allows you to rearrange equations to solve for new variables.
In solving for inverses, you'll often need to:

• Swap the variables involved (typically from y to x or vice versa)
• Eliminate fractions by multiplying through by denominators
• Rearrange terms to isolate the variable you want to solve for

Practicing algebraic manipulation makes complex expressions manageable and aids in deriving functional inverses effectively.

Solving Equations

Solving equations is the art of finding the value of unknowns that make an equation true. It often involves rearranging and simplifying equations to isolate variables.
When deriving the inverse of a function, you're tasked with finding what y equals in terms of x. This often means solving an equation after you switch the roles of these variables.
Steps in solving such equations might include:

• Clearing fractions by multiplying both sides of the equation
• Gathering like terms together
• Isolating the variable on one side of the equation
• Performing operations like addition, subtraction, multiplication, or division to both sides as needed

Mastering these steps helps in easily finding the inverse, ensuring the solutions are functional and correct.

Notation in Mathematics

Mathematical notation is a symbolic way to represent mathematical ideas and objects. It's a language of its own that helps communicate complex concepts in a shorthand and universally understood manner.
For functions, notation often includes:

• Function notation: using symbols like \(f(x)\) to denote a function
• Inverse function notation: represented often as \(f^{-1}(x)\), indicating the function that 'undoes' \(f(x)\)
• Variable swapping in inverse calculations, adjusting x and y roles

Understanding notation is crucial, as it provides a blueprint for accurately expressing and manipulating mathematical functions. It ensures clarity and precision in communicating mathematical ideas, making problems less prone to error.