Problem 18 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Answer: The inverse of the given one-to-one function is $$f^{-1}(x) = \left(\frac{1}{x}\right)^2$$.

Step 1: Replace f(x) with y

Write the function using y instead of f(x): $$y = \frac{1}{\sqrt{x}}$$

Step 2: Swap x and y

Interchange x and y in the equation: $$x = \frac{1}{\sqrt{y}}$$

Step 3: Solve for y

To solve for y, let's first get rid of the denominator by cross-multiplying: $$x\sqrt{y} = 1$$ Now, divide both sides by x: $$\sqrt{y} = \frac{1}{x}$$ Finally, square both sides to isolate y: $$y = \left(\frac{1}{x}\right)^2$$

Step 4: Replace y with the inverse function notation

Replace y with $$f^{-1}(x)$$ to denote the inverse function: $$f^{-1}(x) = \left(\frac{1}{x}\right)^2$$ Thus, the inverse of the given one-to-one function $$f(x) = \frac{1}{\sqrt{x}}$$ is $$f^{-1}(x) = \left(\frac{1}{x}\right)^2$$.

Key Concepts

One-to-One Functions

A one-to-one function is a specific type of function where each output value (or y-value) pairs with exactly one input value (or x-value) and vice versa. In simpler terms, no two different inputs can produce the same output. This characteristic ensures that each y-value is unique to a corresponding x-value, making it possible to reverse the function.

• Ensures unique input-output relationship
• Necessary for finding inverses

One way to check if a function is one-to-one is by using the horizontal line test. If no horizontal line intersects the graph more than once, the function is one-to-one. Understanding this property is crucial when working with inverses, as only one-to-one functions have proper inverse functions.

Solving Equations

Solving equations is a fundamental concept in algebra. It involves finding the value of the unknown variable that makes an equation true. When determining the inverse of a function, you often need to swap variables and solve for the new dependent variable.

• Substitute known values to verify solutions
• Techniques include isolating variables and rearranging terms

In the exercise, after swapping the variables, the main goal of solving for the new y is to use algebraic methods like cross-multiplication or squaring both sides. Understanding these techniques allows you to manipulate and solve equations confidently.

Function Notation

Function notation is a way to express operations in mathematics, often using symbols such as $f(x)$. This format highlights the relationship between inputs (x) and outputs (f(x)). Instead of using generic y, function notation offers a clearer way to denote specific functions and their inverse counterparts.

• Simplifies communication of relationships
• Denotes inverse functions with $f^{-1}(x)$

When transitioning from one function to its inverse, it's crucial to replace y with $f^{-1}(x)$ to signify the function's transformation into its inverse. It’s not just a simple renaming; it represents an entire shift in how inputs and outputs relate to each other.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions to solve equations or find inverses. This often includes operations like adding, subtracting, multiplying, dividing, and even squaring terms to isolate variables.

• Rearrange equations to solve for a specific variable
• Essential for isolating terms to find inverses

In our exercise, after switching variables, it involved multiplying to clear the denominator, and then squaring to eliminate the square root. Mastering these manipulations is crucial for effectively finding the inverse of one-to-one functions. Algebraic manipulation helps you reach the desired solution by systematically changing the structure of equations.