Problem 53 - College Algebra with Modeling and Visualization

Question

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{1}{2}(4-5 x)+1 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the inverse function, denoted as \(f^{-1}(x)\), of the given function \(f(x) = \frac{1}{2}(4 - 5x) + 1\). This involves solving \(y = f(x)\) for \(x\), and then expressing \(x\) in terms of \(y\).

Step 2: Express the Function Clearly

First, express the function more clearly as a single equation: \[ f(x) = \frac{1}{2}(4 - 5x) + 1 = 2 - \frac{5}{2}x \] So, \(f(x) = 2 - \frac{5}{2}x\).

Step 3: Replace f(x) with y

Set \(y = 2 - \frac{5}{2}x\). This means our original function \(f(x)\) is now rewritten as \(y = f(x)\).

Step 4: Solve for x in Terms of y

Rearrange the equation to solve for \(x\):\[ y = 2 - \frac{5}{2}x \] Subtract 2 from both sides to get \[ y - 2 = -\frac{5}{2}x \] Multiply both sides by \(-\frac{2}{5}\) to solve for \(x\):\[ x = \frac{2}{5}(2 - y) \]

Step 5: Express the Inverse Function

Replace \(y\) with \(x\) in the previous equation. Therefore, the inverse function is \(f^{-1}(x) = \frac{2}{5}(2 - x)\).

Key Concepts

Function Notation

Function notation is a useful way to express functions clearly and efficiently. When we talk about functions in math, we usually mean a rule that takes an input, performs some steps, and produces an output.
In our exercise, we used function notation to represent the given function as \( f(x) = \frac{1}{2}(4 - 5x) + 1 \). Here, \( f(x) \) indicates the function applied to the input \( x \), resulting in an output. The purpose of function notation is to simplify expressions and clearly indicate which variable is being transformed.

• "\( f(x) \)" symbolizes the function of \( x \).
• "\( f^{-1}(x) \)" symbolizes the inverse function of \( f(x) \). It essentially reverses what \( f(x) \) does.

Thus, using appropriate notation allows us to easily manipulate and understand the behavior of functions through algebraic processes.

Algebraic Manipulation

Algebraic manipulation is all about rearranging equations to isolate specific variables or simplify expressions. In this problem, we began with \( f(x) = \frac{1}{2}(4 - 5x) + 1 \). Our aim is to manipulate this expression to find the inverse function.
One helpful method is to first simplify the given function, as was done by rewriting it as \( f(x) = 2 - \frac{5}{2}x \). This step makes the expression cleaner and easier to work with.

• Simplify complex expressions to single equations when possible.
• Use basic operations like addition, subtraction, multiplication, and division to rearrange equations.
• Always perform operations equally on all sides of the equation.

Algebraic manipulation forms the core of solving for inverse functions, and by mastering it, one can handle a wide array of algebraic problems efficiently.

Solving Equations

Solving equations is a process of finding the value of the variables that make the equation true. In the context of finding inverse functions, it involves expressing one variable in terms of another.
In our example, we began with \( y = 2 - \frac{5}{2}x \) and worked to express \( x \) in terms of \( y \). Step by step:

• Subtract 2 from both sides to get \( y - 2 = -\frac{5}{2}x \).
• Multiply both sides by \(-\frac{2}{5}\) to isolate \( x \): \( x = \frac{2}{5}(2 - y) \).

Finally, we swapped \( y \) with \( x \) to find the inverse function: \( f^{-1}(x) = \frac{2}{5}(2 - x) \).
These solution steps illustrate the method of systematically applying operations to transform an equation. Solving equations is a fundamental skill that allows us to navigate and resolve mathematical challenges with confidence.