Problem 38 - Elementary and Intermediate Algebra

Question

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. \(f(x)=\frac{x}{3}-\frac{1}{3}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the function definition

The given function is \(f(x) = \frac{x}{3} - \frac{1}{3}\). This function takes an input \(x\), divides it by 3, and then subtracts \(\frac{1}{3}\).

Step 2: Replace f(x) with y

To find the inverse, we start by rewriting the function using \(y\) instead of \(f(x)\): \(y = \frac{x}{3} - \frac{1}{3}\).

Step 3: Solve for x in terms of y

The goal is to express \(x\) in terms of \(y\). Start by adding \(\frac{1}{3}\) to both sides: \(y + \frac{1}{3} = \frac{x}{3}\). Now, multiply both sides by 3 to isolate \(x\): \(3(y + \frac{1}{3}) = x\). Simplify to get \(x = 3y + 1\).

Step 4: Write the inverse function

Now that we have \(x = 3y + 1\), we can express the inverse function by swapping \(x\) and \(y\). The inverse function is \(f^{-1}(x) = 3x + 1\).

Key Concepts

One-to-One Function

To understand what a one-to-one function is, imagine a function as a special kind of machine. This machine takes an input from one set, transforms it, and produces a unique output in another set.
In a one-to-one function, every input has a distinct output, meaning no two different inputs produce the same output.
This uniqueness in mapping ensures that the function can have an inverse. If a function was not one-to-one, two different inputs could yield the same output, messing up any attempt to reverse the process and find the original input given an output. That's why only one-to-one functions can have inverses.
Graphically, you can check if a function is one-to-one using the Horizontal Line Test. If no horizontal line intersects the graph of the function more than once, the function is one-to-one.

Function Notation

Function notation is a way to denote functions in a compact and precise manner.
Instead of saying, "the function of x," we use notation like \( f(x) \). This tells us that \( f \) is the function, and \( x \) is the variable input.Using this notation makes it easier to deal with complex expressions without writing too much. It's like shorthand for math!When finding the inverse of a function, we use \( f^{-1}(x) \).
Keep in mind that \( f^{-1} \) does not mean \( \frac{1}{f} \); instead, it represents the inverse function. This function, when composed with the original function, will return the input value. For instance, if you plug \( f^{-1} \) after \( f \), you'll end up back at the starting point: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).
This highlights the idea that inverse functions 'undo' the action of the original function.

Solving Equations

Equations are like puzzles that require finding the value of a variable that makes the equation true.
Solving an equation involves isolating the variable on one side of the equation.For instance, if you have an equation \( y = \frac{x}{3} - \frac{1}{3} \), and you want to find \( x \) in terms of \( y \), follow these steps:

• Add \( \frac{1}{3} \) to both sides to eliminate the constant on the right. This gives you: \( y + \frac{1}{3} = \frac{x}{3} \).
• Next, multiply both sides by 3 to undo the division. Now you should have \( x = 3(y + \frac{1}{3}) \).
• Finally, simplify the expression to neatly express \( x \) in terms of \( y \), resulting in \( x = 3y + 1 \).

By doing this, you've effectively 'solved' the equation for \( x \). These steps are similar to how you would find an inverse function since you're trying to express one variable in terms of the other.