Problem 12 - College Algebra

Question

For the following exercises, find \(f^{-1}(x)\) for each function. $$ f(x)=\frac{2 x+3}{5 x+4} $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{3 - 4x}{5x - 2}\).

Step 1: Write the equation for inverse

To find the inverse of the function, start by replacing \( f(x) \) with \( y \). So our equation becomes: \( y = \frac{2x + 3}{5x + 4} \).

Step 2: Switch x and y and solve for y

Interchange the roles of \( x \) and \( y \) to find the inverse. The equation now looks like: \( x = \frac{2y + 3}{5y + 4} \). Our goal is to solve for \( y \).

Step 3: Eliminate the fraction

Multiply both sides of the equation by \( 5y + 4 \) to eliminate the fraction: \( x(5y + 4) = 2y + 3 \).

Step 4: Distribute and rearrange terms

Distribute \( x \) on the left side: \( 5xy + 4x = 2y + 3 \). Move all terms involving \( y \) to one side: \( 5xy - 2y = 3 - 4x \).

Step 5: Factor y and solve for y

Factor out \( y \) from the expression: \( y(5x - 2) = 3 - 4x \). Finally, divide by \( (5x - 2) \) to solve for \( y \): \( y = \frac{3 - 4x}{5x - 2} \). This is our inverse function.

Key Concepts

Function Transformation

Understanding function transformations is crucial for grasping inverse functions. When you find the inverse of a function, you are essentially performing a transformation, switching dependent and independent variables. This process involves:

• Reflecting over the line : Inverse functions reflect the original graph over the line \( y = x \).
• Switching x and y : To achieve this reflection, interchange \( x \) and \( y \) in the equation.
• Solve for the new y : Your goal is to express the new equation in terms of \( y \), which represents the inverse.

By understanding these transformations, the process of finding inverse functions becomes much clearer.
Additionally, knowing about function transformations helps in visualizing and understanding more complex operations involving functions.

Rational Functions

At the heart of this exercise lies a rational function, expressed as the ratio of two polynomials. The given function, \( f(x) = \frac{2x + 3}{5x + 4} \), is a classic example. Rational functions, known for their points of discontinuity, often appear in algebraic applications.
Rational functions have key characteristics, including:

• Asymptotes : These are lines that the graph of the function approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal asymptotes depend on the degree of the polynomials involved.
• Domain restrictions : Since the denominator cannot be zero, rational functions have domain limits.

Understanding these properties aids in interpreting and manipulating such functions, particularly when working with their inverses or transformations.

College Algebra

The principles of inverse functions, transformations, and rational functions are part of the core curriculum in college algebra. College algebra provides the foundation for advanced mathematical study and application in various fields.
When dealing with a problem like finding an inverse, you apply skills developed through practice in:

• Algebraic manipulation : Rearranging equations is a key skill, crucial for isolating variables and solving for inverses.
• Critical thinking : Deductive reasoning helps in understanding and implementing the solution steps systematically.
• Graphical analysis : Visualizing functions and their transformations enhances comprehension beyond numerical calculations.

These aspects of college algebra create a versatile toolkit useful beyond academia, in solving real-world problems and advancing in fields that require analytical thinking.