Problem 55 - College Algebra

Question

For the following exercises, find the inverse of the functions with \(a, b, c\) positive real numbers. $$f(x)=\frac{a x+b}{x+c}$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{b - xc}{x - a} \).

Step 1: Replace f(x) with y

The first step in finding the inverse of the function is to replace \( f(x) \) with \( y \). So we rewrite the equation as \( y = \frac{ax + b}{x + c} \).

Step 2: Swap x and y

To find the inverse, swap \( x \) and \( y \) in the equation. This step gives us \( x = \frac{ay + b}{y + c} \).

Step 3: Rearrange to Isolate y

Multiply both sides by \( y + c \) to eliminate the fraction, resulting in \( x(y + c) = ay + b \). Expand to get \( xy + xc = ay + b \).

Step 4: Collect Y Terms on One Side

Move all terms involving \( y \) to one side and constant terms to the other side. This gives \( xy - ay = b - xc \).

Step 5: Factor Out y

Factor \( y \) out of the left-hand side: \( y(x - a) = b - xc \).

Step 6: Solve for y

Divide both sides by \( x - a \) to isolate \( y \): \( y = \frac{b - xc}{x - a} \). This equation represents the inverse function.

Key Concepts

College Algebra

College Algebra serves as a foundation for many mathematical concepts that are expanded upon in various disciplines. It often involves working with functions, which are fundamental sets of operations applied to variables. Functions can be seen as rules that assign each input exactly one output.
Understanding inverse functions in this context is crucial for grasping more complex mathematical ideas. An inverse function effectively "undoes" the operation of the original function. This means that if you apply a function to an input and then apply its inverse to the resulting output, you return to the initial input.

• The notation for an inverse function is usually denoted as \( f^{-1}(x) \).
• Not all functions have inverses, as functions need to be both one-to-one and onto to have an inverse.
• Finding an inverse function involves a series of algebraic manipulations, typically involving swapping variables and solving for the new output variable.

Understanding these operations is a key objective in College Algebra. This prepares students not only for further studies in mathematics but also provides analytical skills applicable in real-world problem solving.

Function Operations

Function operations are a cornerstone of algebra, involving enhancing understanding of how different functions interact. These operations include addition, subtraction, multiplication, and division, as well as composition of functions. Each operation can significantly change the structure and properties of the result.
When dealing with inverse functions, as in the case of our exercise, we are specifically concerned with the composition of functions. Composition is when you apply one function to the results of another function. For the original function \( f(x) = \frac{a x + b}{x + c} \), finding its inverse involves calculating the reverse process, switching the input and output.

• To verify your inverse, compute \( f(f^{-1}(x)) \) or \( f^{-1}(f(x)) \) and ensure it simplifies to \( x \).
• Inverse functions effectively cancel out the operations of the original.

Understanding function operations leads to better manipulation and comprehension of different mathematical models and equations, enabling more precise solutions to complex equations.

Solving Equations

Solving equations is a central skill in algebra that involves finding the values of variables that satisfy the equation. When solving for inverse functions, like in this example, it involves several steps of rewriting and rearranging the equation.
In our example, you initially have the function expressed with variables \(y\) and \(x\). To find the inverse, you:

• Replace \(f(x)\) with \(y\).
• Swap \(x\) and \(y\), fundamentally changing the perspective of the equation.
• Use algebraic manipulation, such as expanding terms and factoring to isolate \(y\) on one side of the equation.

These steps are reflective of systematic problem-solving strategies in algebra. They help cut through complexity to arrive at the inverse, \(y = \frac{b - xc}{x - a}\). Effective equation solving not only pertains to classwork but is also invaluable in scientific research, engineering calculations, and everyday analytics, where outcomes depend on precise and logical conclusions.