Problem 69 - Algebra 2

Question

What is the inverse of \(y=5 x-1 ?\) $$ \begin{array}{lllll}{\text { A. } y=5 x+1} & {\text { B. } y=\frac{x+1}{5}} & {\text { C. } y=\frac{x}{5}+1} & {\text { D. } y=\frac{x}{5}-1}\end{array} $$

Step-by-Step Solution

Answer

The Inverse of the function is \(y=\frac{x+1}{5}\), thus option B is the correct answer.

Step 1: Substitute

Firstly, interchange \(x\) and \(y\) in the equation to give \(x=5y -1\). The purpose of this step is to facilitate the later isolation of \(y\).

Step 2: Isolate \(y\)

Next, get \(y\) on one side of the equation by themselves. To do this, add 1 to both sides of the equation from step 1, and then divide by 5. This gives: \(y=\frac{x+1}{5}\). This is the inverse of the original function.

Step 3: Match the result

Finally, compare the result to the provided options. The answer \(y=\frac{x+1}{5}\) matches option B.

Key Concepts

Function Inversion

Function inversion is the process of finding a function that reverses the effect of the original function. This is akin to finding a way back to the starting point of a mathematical journey. Understanding function inversion is essential in algebra, as it allows you to retrace the steps of a function. Here's how the process works for the given exercise:

• **Substitution:** Replace the roles of the variables. In the exercise, switching the roles of \(x\) and \(y\) brings forth a new perspective.
• **Isolation:** Eliminate other terms to single out the desired variable, which often involves basic algebraic manipulations.

By inverting a function, you create a new function that is the mirror image of the original one on a graph's diagonal line \(y = x\). This concept is especially useful in real-world scenarios, such as converting temperatures from Celsius to Fahrenheit and vice versa. Function inversion involves understanding both the operational structure of the function and how to manipulate algebraic expressions to reveal the new function.

Linear Equations

Linear equations form the backbone of algebra, representing simple yet powerful mathematical expressions that can model a wide array of real-world situations. A linear equation in the context of function inversion maintains a consistent linear relationship between variables.

For example, the original equation \(y = 5x - 1\) is linear because it involves no exponents other than one. Inverting functions poses the task of isolating the dependent variable, leading to a new linear format. In the exercise's context, this results in the inverted form \(y = \frac{x+1}{5}\).

Linear equations are easy to visualize on a graph as straight lines. This straightforward visual representation provides an excellent way to understand the relationship between variables in terms of constant change, making linear equations indispensable in problem-solving.

Mathematical Problem Solving

Mathematical problem solving requires thoughtful application of strategies to arrive at a correct solution. It is the bridge between understanding mathematical theories and applying them to solve problems.

With the given exercise, strategic steps are key in solving for the inverse function.

• **Understanding the Problem:** The description of the problem is crucial. Recognizing that you need the inverse sets the path forward.
• **Choosing an Effective Strategy:** For inversion, swapping variables and isolating the new dependent variable is effective.
• **Verification:** Always verify your answer to ensure it fits with the given choices, as in the exercise where **option B** was confirmed as correct.

Effective mathematical problem solving is much like playing chess with numbers. You must plan moves ahead, anticipate the consequences of transformations, and verify each step for precision.