Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{x+3}{x+7} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{3 - 7x}{x - 1} \).

Step 1: Understand the Function

We are given the function \( f(x) = \frac{x+3}{x+7} \). The goal is to find the inverse, which we will denote as \( f^{-1}(x) \). To find the inverse, we need to solve for \( x \) in terms of \( y \).

Step 2: Replace Variables

Replace \( f(x) \) with \( y \), so the function becomes \( y = \frac{x+3}{x+7} \). Now, solve for \( x \) in terms of \( y \).

Step 3: Cross-Multiply

Multiply both sides of the equation by \( x + 7 \) to clear the fraction: \( y(x + 7) = x + 3 \).

Step 4: Distribute and Rearrange

Distribute \( y \) to get \( yx + 7y = x + 3 \). Rearrange the terms to isolate \( x \) terms on one side and constant terms on the other: \( yx - x = 3 - 7y \).

Step 5: Factor Out x

Factor \( x \) from the left-hand side: \( x(y - 1) = 3 - 7y \).

Step 6: Solve for x

Divide both sides by \( y - 1 \) to solve for \( x \): \( x = \frac{3 - 7y}{y - 1} \).

Step 7: Express Inverse Function

Now express \( x \) as \( f^{-1}(y) \), effectively replacing \( y \) back with \( x \): \( f^{-1}(x) = \frac{3 - 7x}{x - 1} \).

Key Concepts

Function Notation

Function notation is a way to name and define functions in mathematics. It is conventionally represented as \( f(x) \), where "\( f \)" denotes the function, and "\( x \)" is the variable or input to the function.
By specifying \( f(x) = \frac{x+3}{x+7} \), we are defining a relationship between inputs and outputs.

• The notation helps us identify which function we are working with and what its input is.
• It acts like a labeling system to keep track of different functions in a problem.

In this particular exercise, we use \( f(x) \) to denote our function initially and later use \( y = f(x) \) to facilitate solving for the inverse. It's a like a shorthand language of mathematics that simplifies communicating complex relationships.

Cross-Multiplication

Cross-multiplication is a powerful technique used to solve equations involving fractions. It helps to clear out fractions by multiplying terms across an equation.
For the function \( y = \frac{x+3}{x+7} \), we apply cross-multiplication to eliminate the fraction:

• Multiply both sides by \( x+7 \) to get \( y(x+7) = x+3 \).
• This ensures we have a fraction-free equation that is easier to manipulate further.

Cross-multiplying simplifies the problem by transforming it into a linear equation, paving way for further operations like distribution and rearranging terms. This makes solving for unknowns more straightforward.

Factorization

Factorization is the process of breaking down expressions into products of simpler expressions or factors. It is a crucial step in solving equations because it allows us to simplify expressions and solve for variables more efficiently.
In the equation \( yx + 7y = x + 3 \), after rearranging, we have:

• Isolate the \( x \) terms: \( yx - x = 3 - 7y \).
• Factor \( x \) out from the left hand side to get \( x(y-1) \).

This gives \( x(y-1) = 3 - 7y \), where \( x \) is now clear of other terms, facilitating solving. By expressing \( x \) in terms of other expressions, factorization streamlines the path to finding our inverse function.

Solving Equations

Solving equations involves finding the value of unknown variables that satisfy the equation. It is a fundamental aspect of algebra and calculus.
In this exercise, after factorization, we need to solve for \( x \):

• Divide both sides of the equation \( x(y-1) = 3 - 7y \) by \( y-1 \), resulting in \( x = \frac{3 - 7y}{y-1} \).
• This step is crucial because it isolates \( x \) completely, expressing it in terms of \( y \).

Finally, replace \( y \) with \( x \) to express the inverse function: \( f^{-1}(x) = \frac{3 - 7x}{x - 1} \). Solving equations like these helps us find our inverse function, a crucial step in understanding the behavior of the original function.