Problem 10 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Definition of an Inverse Function

An inverse function, denoted as \(f^{-1}(x)\), reverses the effect of the original function \(f(x)\). This means that if \(f(a) = b\), then \(f^{-1}(b) = a\).

Step 2: Set Up the Equation

Set the original function equal to \(y\). This helps in forming an equation that will be solved for \(x\): \ \[ y = 3 - x \]

Step 3: Solve for x

Solve the equation for \(x\) in terms of \(y\). This is done by isolating \(x\) on one side of the equation: \ Add \(x\) to both sides: \ \[ y + x = 3 \] Subtract \(y\) from both sides: \ \[ x = 3 - y \]

Step 4: Express the Inverse Function

Replace \(y\) with \(x\) in the expression obtained for \(x\) to write the inverse function. The inverse function \(f^{-1}(x)\) is obtained as follows: \ \[ f^{-1}(x) = 3 - x \]

Key Concepts

Function Notation

Function notation is an essential concept in mathematics, as it provides a formal way to describe the process of mapping elements from one set (domain) to another (range). This notation utilizes symbols like \( f(x) \) to denote a function named \( f \) with an input variable \( x \). It's like labeling a machine through which inputs pass to get transformed into outputs. For instance, when we write \( f(x) = 3 - x \), it tells us that the function \( f \) takes any number \( x \) and transforms it by subtracting \( x \) from 3.
Function notation is not just about dealing with numbers; it lays the groundwork for expressing various mathematical operations clearly.

• It is used to symbolize the output that is a result of processing the input through the function rules.
• This notation also makes it easy to work with composite functions and inverses.

By understanding function notation, students can grasp more complex mathematical concepts with ease.

Solving Equations

Solving equations is pivotal in finding unknown values in mathematical problems. An equation is a statement that asserts the equality of two expressions. In the context of inverse functions, solving equations helps us in understanding the relationship between the function and its inverse.
To find an inverse function, begin by treating the function as an equation. For the function \( f(x) = 3 - x \), set it equal to \( y \), getting \( y = 3 - x \).

• First, we aim to isolate the variable \( x \) in terms of \( y \).
• To isolate \( x \), the equation can be rearranged in simple algebraic steps: Add \( x \) to both sides, resulting in \( y + x = 3 \), then subtract \( y \) from both sides to get \( x = 3 - y \).

Solving these equations involves rearranging them to isolate the desired variable. This leads us directly to the formula for the inverse function.

Inverse Operations

Inverse operations are operations that undo each other, similar to how an inverse function undoes the effect of the original function. These operations are foundational in solving equations and understanding inverses.
For the given function \( f(x) = 3 - x \), to reverse this operation, we switch the roles of \( x \) and \( y \) and solve for the new \( y \) (which originally was \( x \)). By solving \( y = 3 - x \) to \( x = 3 - y \), we effectively apply inverse operations to derive \( f^{-1}(x) = 3 - x \).

• Inverse operations in mathematics often involve switching addition with subtraction or multiplication with division.
• Understanding these operations is crucial because they help simplify complex problems by dismantling them back to their simplest form.

Grasping inverse operations equips students to tackle inverse functions with ease, revealing the hidden relationships within given equations.