Problem 18 - College Algebra

Question

For the following exercises, find the inverse of the functions. $$ f(x)=9+\sqrt{4 x-4} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{(x - 9)^2 + 4}{4} \).

Step 1: Understand the Function

The given function is \( f(x) = 9 + \sqrt{4x - 4} \). To find the inverse, we need to understand that we will swap the roles of \(x\) and \(y\), where \(y = f(x)\).

Step 2: Set Up the Equation for Inverse

Rewrite the function as \( y = 9 + \sqrt{4x - 4} \). To find the inverse, solve for \(x\) in terms of \(y\): \( y = 9 + \sqrt{4x - 4} \).

Step 3: Isolate the Square Root

Subtract 9 from both sides to isolate the square root: \( y - 9 = \sqrt{4x - 4} \).

Step 4: Square Both Sides

Square both sides to remove the square root: \((y - 9)^2 = 4x - 4\).

Step 5: Solve for x

Add 4 to both sides to get \((y - 9)^2 + 4 = 4x\). Then divide by 4: \( x = \frac{(y - 9)^2 + 4}{4} \). This expresses \(x\) in terms of \(y\).

Step 6: Express the Inverse Function

Swap \(x\) and \(y\) to find the inverse function: \( f^{-1}(x) = \frac{(x - 9)^2 + 4}{4} \).

Key Concepts

function notation

In mathematics, function notation is a way to represent functions in a clear and concise manner. It typically involves letters such as \( f, g, \text{ or } h \) and uses the format \( f(x) \), where \( f \) represents the function, and \( x \) represents the input to the function. For example, in the function \( f(x) = 9 + \sqrt{4x - 4} \), \( f \) is the name of the function, and \( x \) is the variable we input into the function to calculate an output.

Function notation helps keep track of different operations and transformations applied to input values. This not only provides a way to discuss functions algebraically but also allows us to explore relationships between variables.

When you see a function, remember that every input "x" has exactly one output "f(x)". In the original exercise, we're actually seeking the inverse function, which means reversing the roles of inputs and outputs to see the function running backward.

solving equations

Solving equations is a fundamental skill in algebra, vital for finding unknown values in mathematical problems. When solving an equation, our goal is to isolate the unknown variable on one side of the equation.

For instance, given the equation \( y = 9 + \sqrt{4x - 4} \), our task is to solve for \( x \) in terms of \( y \). This involves several algebraic steps, each designed to gradually simplify the equation:

• First, subtract 9 from both sides to help isolate the variable expression with the square root: \( y - 9 = \sqrt{4x - 4} \).
• Then, square both sides of the equation to eliminate the square root: \((y - 9)^2 = 4x - 4\).
• Finally, solve for \( x \) by further manipulating the equation: add 4 and divide by 4.

These steps illustrate how solving equations is an incremental process, breaking the problem down into simpler operations to uncover the unknown variable.

square roots

Square roots are an important concept in mathematics, signifying the operation of finding a number that, when multiplied by itself, gives the original number. In this context, it is denoted by the symbol \( \sqrt{\cdot} \).

In the function \( f(x) = 9 + \sqrt{4x - 4} \), the square root part, \( \sqrt{4x - 4} \), is a pivotal aspect of the equation.

The process of inverse function finding often requires removing the square root to simplify the equation. One common technique is squaring both sides of the equation, which neutralizes the square root and allows further algebraic manipulation.

Remember, when handling square roots:

• Ensure expression under the square root is non-negative (since we're dealing with real numbers).
• Be cautious with sign changes when squaring both sides to eliminate the root.

This practice is crucial in many areas of mathematics and science, as it often simplifies the calculation of an inverse function or solving equations.

algebraic manipulation

Algebraic manipulation refers to using various algebraic techniques to simplify or rearrange equations or expressions. This skill is foundational in solving mathematical problems, including finding inverse functions.

In the provided problem, we perform several algebraic manipulations to find the inverse of the function \( f(x) = 9 + \sqrt{4x - 4} \). Here’s a breakdown of some techniques involved:

• Isolating terms: Subtract and add numbers to keep key terms by themselves, such as subtracting 9 in \( y - 9 = \sqrt{4x - 4} \).
• Undo operations: Use inverse operations, like squaring to remove a square root, demonstrated in \( (y - 9)^2 = 4x - 4 \).
• Rearranging: Adjust the equation structure to solve for the desired variable, culminating in \( x = \frac{(y - 9)^2 + 4}{4} \).

These manipulations transform a complex expression into a more accessible one, enabling easier and more efficient problem-solving in mathematics and real-world applications.