Problem 51 - Algebra and Trigonometry

Question

Find the inverse function of \(f\) $$ f(x)=\sqrt{2+5 x} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{x^2 - 2}{5} \).

Step 1: Understand the Function

The function given is \( f(x) = \sqrt{2 + 5x} \). Our task is to find its inverse.

Step 2: Replace f(x) with y

To find the inverse, we start by replacing \( f(x) \) with \( y \) to have the equation \( y = \sqrt{2 + 5x} \).

Step 3: Swap x and y

To find the inverse function, we swap the roles of \( x \) and \( y \). This gives us the equation \( x = \sqrt{2 + 5y} \).

Step 4: Solve for y

To isolate \( y \), first square both sides to eliminate the square root: \( x^2 = 2 + 5y \).

Step 5: Isolate y

Rearrange the equation to solve for \( y \): \( 5y = x^2 - 2 \). Finally, divide by 5 to get \( y = \frac{x^2 - 2}{5} \).

Step 6: Conclusion: Write the Inverse Function

The inverse function of \( f(x) \) is \( f^{-1}(x) = \frac{x^2 - 2}{5} \).

Key Concepts

Function Transformation

Understanding function transformation is essential when dealing with inverse functions. It involves manipulating the given function into a different form by applying specific mathematical operations. In our exercise, the original function undergoes several transformations to find its inverse.

• Initially, the function is expressed in the standard form, \( f(x) = \sqrt{2 + 5x} \).
• The first important transformation occurs when we replace \( f(x) \) with \( y \), setting the stage for inverse operations.
• The next transformation involves swapping \( x \) and \( y \), indicating where the inverse function will take input and output. The equation becomes \( x = \sqrt{2 + 5y} \).
• Finally, the logical use of algebraic manipulation to solve for \( y \), such as squaring the equation to get \( x^2 = 2 + 5y \) and isolating \( y \), leads to the inverse function’s expression.

Through these transformations, we effectively reverse the operations of the original function.

Square Root Function

The square root function is a vital component in our exercise. Understanding its properties allows us to find the inverse accurately. The function is \( f(x) = \sqrt{2 + 5x} \) and represents the square root of a specific linear expression.

• This type of function only takes inputs that result in non-negative values because the square root of a negative number is not defined in the real number system.
• The domain of the square root function \( f(x) \) affects the range of its inverse, which is important in ensuring that the inverse function maps correctly.
• Squaring both sides when manipulating the equation is crucial for eliminating the square root, simplifying further steps.

Grasping these aspects of square root functions enables us to solve for the inverse efficiently.

Solving Equations

Solving equations is a central part of finding inverse functions. To derive the inverse function of \( f(x) = \sqrt{2 + 5x} \), we require systematic algebraic processes.

• The first key move is to swap \( x \) and \( y \), which is a conceptual shift necessary for finding an inverse. It leads to the equation \( x = \sqrt{2 + 5y} \).
• To remove the square root, square the equation, resulting in \( x^2 = 2 + 5y \). This step requires caution as it alters the domain and range considerations of the original equation.
• Rearrange the equation to isolate and solve for \( y \), resulting in \( 5y = x^2 - 2 \).
• Finally, divide by 5 to completely isolate \( y \), yielding the inverse function \( y = \frac{x^2 - 2}{5} \).

Being meticulous in each step ensures we solve for the desired inverse accurately.

Function Notation

Function notation is a helpful tool used to describe functions clearly and specify their components, like inputs and outputs. In this exercise, it sets a framework for better understanding the relationships between the original function and its inverse.

• The function is originally given as \( f(x) = \sqrt{2 + 5x} \), where \( f(x) \) signals the output corresponding to the input \( x \).
• By converting \( f(x) \) to \( y \), we simplify the process of finding the inverse because it aligns with conventional equation-solving methods.
• Finally, function notation helps in conclusively presenting the inverse as \( f^{-1}(x) = \frac{x^2 - 2}{5} \), a clear identifier of the inverse relationship to \( f(x) \).

Mastering function notation enables us to track transformations and results clearly, contributing to understanding and solving inverse function problems.