Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{3}{x-4} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{3+4x}{x} \).

Step 1: Replace f(x) with y

Start by replacing \( f(x) \) with \( y \) in the equation. This gives us:\[ y = \frac{3}{x-4} \]

Step 2: Swap x and y

To find the inverse, switch the variables \( x \) and \( y \). Now, solve in terms of \( y \):\[ x = \frac{3}{y-4} \]

Step 3: Solve for y

Our goal is to isolate \( y \) on one side of the equation. Start by multiplying both sides by \( y - 4 \):\[ x(y - 4) = 3 \]Next, distribute \( x \):\[ xy - 4x = 3 \]Add \( 4x \) to both sides:\[ xy = 3 + 4x \]Then, divide both sides by \( x \):\[ y = \frac{3 + 4x}{x} \]

Step 4: Notation for Inverse Function

The inverse function of \( f \), denoted \( f^{-1}(x) \), is written as:\[ f^{-1}(x) = \frac{3+4x}{x} \]

Key Concepts

Function Notation

Function notation is a way to name and evaluate functions clearly and concisely. In mathematics, a function is often represented by symbols like \( f(x) \), which tells us that the function's name is \( f \) and \( x \) is the variable being used. The expression inside the parentheses, such as \( x \), represents the input value that is plugged into the function formula.

When you see an equation written as \( f(x) = \frac{3}{x-4} \), it means the output of the function depends on what is substituted for \( x \). Function notation allows us to easily substitute different values into the function to calculate corresponding outputs. For example, if \( x = 5 \), substitute 5 into the equation to find \( f(5) \).

• Substitute \( x \) with 5: \( f(5) = \frac{3}{5-4} = 3 \)
• This tells us that when \( x = 5 \), the output, or the value of \( f \), is 3.

Understanding function notation is essential for studying more advanced topics in mathematics, such as derivatives and integrals, as it provides a standardized way of expressing and manipulating functions.

Solving Equations

Solving equations is a fundamental skill in mathematics. It involves finding the value of variables that satisfy an equation. In the context of finding inverse functions, it means solving for the variable after switching the function’s variables.

For our function \( f(x) = \frac{3}{x-4} \), once we replace \( f(x) \) with \( y \), the task is to find the inverse by solving the equation when we switch the variables to \( x = \frac{3}{y-4} \). Solving equations generally involves:

• Isolating the variable of interest, in this case \( y \).
• Using operations like adding, subtracting, multiplying, and dividing both sides of the equation to achieve this.

In our example:

• We multiply both sides by \( y - 4 \) to eliminate the denominator: \( x(y-4) = 3 \).
• Distribute \( x \) over \( y - 4 \) to get \( xy - 4x = 3 \).
• Add \( 4x \) to both sides: \( xy = 3 + 4x \).
• Finally, divide both sides by \( x \) to solve for \( y \): \( y = \frac{3 + 4x}{x} \).

These steps allow us to arrive at the solution for the inverse function, illustrating the importance of equation-solving skills.

Inverse Notation

Inverse notation is an essential part of function theory. When we talk about the inverse of a function \( f \), we denote it as \( f^{-1} \). This indicates that \( f^{-1} \) is a function that "undoes" the effect of \( f \). In simple terms, if \( f(x) \) takes an input \( x \) to produce an output \( y \), then \( f^{-1}(y) \) takes \( y \) back to \( x \).

Inverse functions interchange the roles of the dependent and independent variables. This means if \( f(x) = \frac{3}{x-4} \), then its inverse, \( f^{-1}(x) \), must be determined by reversing the operations used in \( f \).

To correctly write a function's inverse, you:

• Swap the roles of \( x \) and \( y \).
• Solve the resulting equation to find \( y \), which gives you the formula for \( f^{-1}(x) \).

In our specific example, hence, once calculated, the inverse notation becomes \( f^{-1}(x) = \frac{3 + 4x}{x} \). This notation clearly expresses that reversing the function is achievable, and it provides valuable insight into how functions interact with one another.