Problem 19 - College Algebra

Question

For the following exercises, find the inverse of the functions. $$ f(x)=\sqrt{6 x-8}+5 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{(x - 5)^2 + 8}{6} \).

Step 1: Understanding the Function

Firstly, the given function is \( f(x) = \sqrt{6x - 8} + 5 \). Our task is to find the inverse function, \( f^{-1}(x) \), such that \( f(f^{-1}(x)) = x \).

Step 2: Rewriting the Function in Terms of y

Let's write the function with \( y \) instead of \( f(x) \): \( y = \sqrt{6x - 8} + 5 \). Our goal is to solve for \( x \) in terms of \( y \).

Step 3: Isolating the Square Root

Subtract 5 from both sides to isolate the square root: \( y - 5 = \sqrt{6x - 8} \).

Step 4: Removing the Square Root

Square both sides to eliminate the square root: \( (y - 5)^2 = 6x - 8 \).

Step 5: Solving for x

Add 8 to both sides: \( (y - 5)^2 + 8 = 6x \). Then divide by 6 to solve for \( x \): \( x = \frac{(y - 5)^2 + 8}{6} \).

Step 6: Expressing the Inverse Function

Replace \( y \) with \( x \) to denote the inverse function: \( f^{-1}(x) = \frac{(x - 5)^2 + 8}{6} \). Thus, the inverse function is \( f^{-1}(x) = \frac{(x - 5)^2 + 8}{6} \).

Key Concepts

Functions

Functions are fundamental components in mathematics. A function, represented as \( f(x) \), is essentially a rule that assigns each input \( x \) to exactly one output. Think of it like a vending machine: you input a specific code (the function's input), and it gives you a particular snack (the output). Functions are often represented with equations. For example, \( f(x) = \sqrt{6x - 8} + 5 \) is a function. Here, each value of \( x \) results in a unique \( f(x) \).
Understanding functions involves knowing:

• Input: The number you provide to the function, usually denoted as \( x \).
• Output: The result you get from the function, often denoted as \( f(x) \) or \( y \).
• Rule: The equation or process applied to the input to determine the output.

The concept of an inverse function arises from reversing this process. If the original function is \( f(x) \), its inverse, \( f^{-1}(x) \), backtracks to find the original input from the output.

Square Root

The square root is one of the mathematical operations used in many functions, like the one we are working with. The square root of a number \( a \), written as \( \sqrt{a} \), is a value that, when multiplied by itself, yields \( a \). For example, \( \sqrt{9} = 3 \), because \( 3 \times 3 = 9 \).
When dealing with the square root in functions, it's important to remember:

• Square roots often have two possible solutions, positive and negative, but in functions, we usually consider the principal (positive) square root.
• This operation can be reversed by squaring, which is useful when finding the inverse of a function.

In our original function \( f(x) = \sqrt{6x - 8} + 5 \), the square root affects how the input \( x \) is processed, complicating the isolation of \( x \) in the inverse function process.

Solving Equations

Solving equations is a key skill in mathematics, involving the determination of unknown values that satisfy a particular condition. For example, in our function problem, we needed to find the inverse of \( f(x) = \sqrt{6x - 8} + 5 \), which meant solving for \( x \) in terms of \( y \).
Here's a handy step-by-step approach to tackle such problems:

• Substitute the Function: Replace \( f(x) \) with \( y \) to simplify the relationship.
• Isolate the Component: Move terms around to isolate the part of the equation involving the unknown. Here, moving \( 5 \) lets us isolate the square root.
• Eliminate Complex Operations: In our example, eliminating the square root by squaring both sides makes solving for \( x \) more straightforward.
• Solve Step-by-Step: Perform algebraic operations to isolate \( x \). Pay attention to operations applied to both sides equally.

By following these steps, you can solve equations and understand inverse functions more easily, helping demystify complex relationships and making the math accessible.