Problem 23 - Algebra and Trigonometry

Question

If \(f(x)=5-2 x,\) find \(f^{-1}(3)\)

Step-by-Step Solution

Answer

The value of \(f^{-1}(3)\) is 1.

Step 1: Understand the Problem

We need to find the inverse function, denoted as \(f^{-1}(x)\), and then evaluate it at \(x = 3\). The given function is \(f(x) = 5 - 2x\).

Step 2: Swap Variables

To find the inverse function, start by writing \(y = f(x) = 5 - 2x\). Now swap \(x\) and \(y\) to get \(x = 5 - 2y\).

Step 3: Solve for y

Solve the equation \(x = 5 - 2y\) for \(y\). First, subtract 5 from both sides: \(x - 5 = -2y\). Then, divide by -2: \(y = \frac{5 - x}{2}\).

Step 4: Write the Inverse Function

The inverse function can now be written as \(f^{-1}(x) = \frac{5 - x}{2}\).

Step 5: Evaluate the Inverse Function at x=3

Substitute \(x = 3\) into \(f^{-1}(x)\): \(f^{-1}(3) = \frac{5 - 3}{2} = \frac{2}{2} = 1\).

Key Concepts

Function Notation

Function notation is a way to represent functions in mathematics. It involves writing functions in terms of input-output relationships using variables like \(x\) and \(y\). For example, in the given function \(f(x) = 5 - 2x\), \(x\) represents the input, and \(f(x)\) symbolizes the output.
This notation helps clarify which variable is independent (\(x\) in this case) and how it relates to the dependent variable (\(f(x)\) or \(y\)). By using function notation, we not only identify the function itself but also set the stage for operations like finding inverses.
An inverse function, denoted as \(f^{-1}(x)\), essentially swaps these roles. If originally \(f(x)\) relates each \(x\) to a \(y\), then \(f^{-1}(x)\) should relate each \(y\) back to an \(x\). Function notation thus provides a straightforward framework for expressing these relationships clearly and accurately.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations to solve for a particular variable. When finding inverse functions, this is crucial. Take the equation \(y = 5 - 2x\) from our original function \(f(x) = 5 - 2x\).
To find the inverse, we first swap \(x\) and \(y\) to get \(x = 5 - 2y\). This swapping echoes the concept that the inverse should reverse the function's effect.
Next, we use algebraic manipulation to solve for \(y\):

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

These steps help in expressing \(y\) solely in terms of \(x\), allowing us to write the inverse as \(f^{-1}(x) = \frac{5 - x}{2}\).
Algebraic manipulation enables us to unravel these inverse relationships, giving us the tools to express one variable in terms of another.

Function Evaluation

Function evaluation is about substituting specific values into a function. Once you've found the inverse function, \(f^{-1}(x) = \frac{5 - x}{2}\), it's time to evaluate it at a given point.
The problem asks us to find \(f^{-1}(3)\). This means we substitute \(3\) wherever we see \(x\) in the inverse function.

• Replace \(x\) with \(3\) in \(f^{-1}(x)\), giving \(f^{-1}(3) = \frac{5 - 3}{2}\).
• Calculate \(f^{-1}(3) = \frac{2}{2} = 1\).

Evaluating a function tells us the output for a specific input.
It’s crucial to correctly substitute and simplify expressions to get the right answers. Here, we determined that the value at \(x=3\) using the inverse is \(1\). This confirms our evaluation process is correct.