Problem 45 - College Algebra with Modeling and Visualization

Question

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=3 x-1 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Given Function

The function given is \( f(x) = 3x - 1 \). This is a linear function, where the transformation involves scaling by 3 and then translating by -1. To find the inverse function \( f^{-1}(x) \), we need to reverse these operations.

Step 2: Replace Function Notation

Start by rewriting the function \( f(x) \) using \( y \) as \( y = 3x - 1 \). This sets up the equation where we will solve for \( x \) in terms of \( y \).

Step 3: Solve for \( x \) in Terms of \( y \)

To isolate \( x \), we first add 1 to both sides: \( y + 1 = 3x \). Then, divide both sides by 3 to solve for \( x \): \( x = \frac{y + 1}{3} \).

Step 4: Swap \( x \) and \( y \) to Find \( f^{-1}(x) \)

The expression for \( x \) we derived in terms of \( y \) becomes the inverse function when we swap \( x \) and \( y \). Thus, the inverse function \( f^{-1}(x) \) is \( f^{-1}(x) = \frac{x + 1}{3} \).

Step 5: Verify the Inverse Function

To ensure correctness, check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) hold. Substitute \( f^{-1}(x) = \frac{x + 1}{3} \) into \( f(x) \) and vice versa. Both should simplify to \( x \). After verifying, both operations satisfy the conditions for inverse functions.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics. It involves finding the value of variables that make an equation true. For instance, in the given function setup, we start with the equation \( y = 3x - 1 \). Here, our task is to solve for \( x \) in terms of \( y \). This helps us find the inverse of the function.

In solving equations, each operation must be precisely reversed to keep the equation balanced. This could involve addition, subtraction, multiplication, or division. The main concept is maintaining equality throughout each step. In our example, we first add 1 to both sides to adjust the equation to \( y + 1 = 3x \). Next, dividing each side by 3 gives us \( x = \frac{y + 1}{3} \). Solving like this sets up the path to find inverse functions. By replacing \( y \) with \( x \) in the final step, we determine the inverse function \( f^{-1}(x) \).

This demonstrates how equations are used to find function inverses by reversing each operation one by one. Mastering these steps opens the door to solving more complex problems.

Linear Functions

Linear functions are one of the simplest types of functions and easily recognizable due to their straight-line graph. A linear function can be expressed generally as \( f(x) = ax + b \), where \( a \) and \( b \) are constants. The given function \( f(x) = 3x - 1 \) is a perfect example, with 3 as the slope and -1 as the y-intercept.

This function describes a straight line where every change in \( x \) by 1 unit causes a change in \( f(x) \) by 3 units. The transformation involves scaling the input by the slope and then translating the result up or down by the intercept. Linear functions are consistent and predictable. They're used in countless applications from computing interest in finance to analyzing trends in data.

Understanding linear functions is crucial when finding inverse functions. By reversing their transformations—such as swapping the roles of \( x \) and \( y \) and solving for the original variable—it becomes straightforward to derive their inverse form. The straightforward nature of linear functions makes them an excellent entry point for students exploring more intricate mathematical concepts.

Function Notation

Function notation is a way of expressing functions in mathematics, commonly written as \( f(x) \). It provides a clear and standardized way to depict the relationship between inputs and outputs. In the given exercise, \( f(x) = 3x - 1 \) specifies a particular output for every input \( x \). This setup paves the way for determining inverses and solutions.

A pivotal step in our task was substituting the function notation \( f(x) \) with \( y \) to align with solving equations for inverse functions. Notation plays a vital role here. It helped transition between expressing relationships in the function and isolating variables when solving.

• Uses standardized symbols to define relationships.
• enables precise substitution and manipulation.
• Facilitates clearer communication in solving problems.

Comprehending function notation allows seamless navigation through different mathematical operations, as it standardizes the language used among different equations and functions. This consistency makes it easier to understand and solve various mathematical problems. By mastering function notation, the process of solving and manipulating equations becomes much more intuitive, benefiting any student aiming to advance their analytical skills.