Problem 16 - College Algebra

Question

For the following exercises, find the inverse of the functions. $$ f(x)=\sqrt{2 x+1} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{x^2 - 1}{2} \).

Step 1: Understand the Concept of Inverse

To find the inverse of a function, we essentially swap the roles of the dependent and independent variables. This involves replacing the function notation with a variable representing the output and then solving for the original variable.

Step 2: Replace Function Notation with Variable

Start by replacing \( f(x) \) with \( y \). This gives us the equation \( y = \sqrt{2x+1} \).

Step 3: Swap Variables

To find the inverse, swap \( x \) and \( y \). This gives \( x = \sqrt{2y+1} \).

Step 4: Solve for the New Output

Solve the equation \( x = \sqrt{2y+1} \) for \( y \). First, square both sides to eliminate the square root: \( x^2 = 2y + 1 \).

Step 5: Isolate the New Output Variable

Rearrange the equation to solve for \( y \): Subtract 1 from both sides to get \( x^2 - 1 = 2y \). Then, divide by 2, resulting in \( y = \frac{x^2 - 1}{2} \).

Step 6: Write the Inverse Function

Rewrite \( y \) as \( f^{-1}(x) \) to denote the inverse function. Therefore, the inverse function is \( f^{-1}(x) = \frac{x^2 - 1}{2} \).

Key Concepts

Function Notation

Function notation is a way of expressing the relationship between inputs and outputs in mathematical terms. It is a shorthand way to refer to functions, using symbols and letters, like \( f(x) \). This notation tells us that the function named \( f \) takes \( x \) as an input.Function notation can become particularly useful when dealing with multiple functions or more complex equations. It allows us to easily reference and manipulate these equations without writing them fully each time.

• Replacing Function Notation: When working with inverse functions, the first step is to replace \( f(x) \) with \( y \). This simplifies the process of solving the equation for the inverse.

Let's consider the function \( f(x) = \sqrt{2x+1} \). By rewriting it as \( y = \sqrt{2x+1} \), we prepare it for the manipulation needed to find its inverse.Function notation not only helps in organizing and solving equations but also plays a crucial role in understanding different function transformations, like finding inverses.

Solving Equations

Solving equations is an important process in finding inverse functions and other mathematical operations. It involves rearranging equations to isolate the desired variable, making it possible to understand the relationship between different elements in the function.The process typically involves:

• Identifying the Equation: Start with the equation you need to solve, which is often rephrased from its original function notation.
• Swapping Variables: For the purpose of finding an inverse, swap the related variables (e.g., replacing \( x \) with \( y \)). This allows us to solve for the original input variable in terms of the output.

For example, given \( y = \sqrt{2x+1} \), swap \( x \) and \( y \) to have \( x = \sqrt{2y+1} \). By solving this new equation, you can derive the inverse function.Solving such equations often requires breaking down complex calculations into simpler steps, ensuring each part of the equation is balanced and correctly manipulated.

Algebraic Manipulation

Algebraic manipulation is the art of adjusting an expression or equation, in such a way, to isolate a specific variable or solve the equation. This process often involves arithmetic operations like addition, subtraction, multiplication, and division, as well as the application of more complex operations like squaring or taking roots.

Steps in Algebraic Manipulation

In the context of our example, \( x = \sqrt{2y+1} \) becomes the starting point for our algebraic manipulation.

• Eliminating Radicals: Square both sides to remove the square root. This gives \( x^2 = 2y + 1 \).
• Rearranging the Equation: Adjust the equation to isolate \( y \). This involves subtracting 1 from both sides, yielding \( x^2 - 1 = 2y \).
• Isolating the Variable: Finally, divide by 2 so that \( y \) stands alone: \( y = \frac{x^2 - 1}{2} \). This final expression represents the inverse function.

By understanding and practicing these types of algebraic manipulations, you can confidently solve for inverses and tackle more complex functions with ease. These skills are not only vital for math problems but are applicable in various real-world situations.