Finding The Inverse Of A Linear Function Step By Step Solution

In the realm of mathematics, functions are fundamental building blocks that describe relationships between variables. Among these, linear functions hold a special place due to their simplicity and wide applicability. Understanding the properties of linear functions, including how to find their inverses, is crucial for various mathematical and real-world applications. This article delves into the process of finding the inverse of a given linear function, $f(x) = \frac{x+3}{2}$, and explores its implications. We will address the following key questions:

• What is the value of $f(2)$?
• How do we determine the inverse function, $f^{-1}(x)$?
• What is the value of the inverse function at $x = 12$, i.e., $f^{-1}(12)$?

By answering these questions, we aim to provide a comprehensive understanding of linear functions and their inverses.

a) Finding $f(2)$: Evaluating the Function

The first step in understanding a function is to evaluate it at specific points. This gives us a sense of how the function transforms input values into output values. In this case, we are asked to find $f(2)$ for the function $f(x) = \frac{x+3}{2}$. This means we need to substitute $x = 2$ into the function and simplify the expression.

Evaluating the function at a specific point involves replacing the variable x with the given value and performing the arithmetic operations. This process allows us to determine the output of the function for a particular input. In the context of linear functions, this evaluation helps us understand the linear relationship between the input and output values. For instance, knowing $f(2)$ tells us the specific output value when the input is 2, providing a concrete point on the line represented by the function.

To find $f(2)$, we substitute $x = 2$ into the expression for $f(x)$:

$f(2) = \frac{2 + 3}{2}$

Now, we simplify the numerator:

$f(2) = \frac{5}{2}$

Therefore, $f(2) = 2.5$. This means that when the input to the function is 2, the output is 2.5. This calculation demonstrates the direct application of the function's definition to find the output for a given input. Understanding this process is fundamental to working with functions and their inverses.

This straightforward calculation highlights the nature of linear functions, where a constant change in the input results in a proportional change in the output. In this case, we have found a specific point on the line defined by $f(x)$, which is crucial for further analysis, such as understanding the function's graph or determining its inverse. Evaluating functions at specific points is a cornerstone of mathematical analysis, providing tangible insights into the behavior and characteristics of the function.

b) Finding $f^{-1}(x)$: Determining the Inverse Function

The concept of an inverse function is crucial in mathematics, particularly when dealing with functions that have a one-to-one correspondence between their input and output values. The inverse function, denoted as $f^{-1}(x)$, essentially "undoes" the operation of the original function $f(x)$. In other words, if $f(a) = b$, then $f^{-1}(b) = a$. Finding the inverse function involves a systematic process of interchanging the roles of the input and output variables and then solving for the new output variable.

To find the inverse function $f^{-1}(x)$, we start by rewriting the original function $f(x) = \frac{x+3}{2}$ as $y = \frac{x+3}{2}$. The next step is to swap the variables $x$ and $y$, which gives us $x = \frac{y+3}{2}$. This step is the core of finding the inverse, as it reflects the reversal of the function's operation. Now, we need to solve this equation for $y$, which will give us the expression for the inverse function.

To solve for $y$, we first multiply both sides of the equation by 2:

$2x = y + 3$

Next, we subtract 3 from both sides:

$2x - 3 = y$

Thus, the inverse function is:

$f^{-1}(x) = 2x - 3$

This equation represents the inverse of the original function. The inverse function takes an input $x$, multiplies it by 2, and then subtracts 3 to produce the output. This process effectively reverses the operations performed by the original function, highlighting the fundamental relationship between a function and its inverse.

The inverse function we found, $f^{-1}(x) = 2x - 3$, is another linear function, which is a common characteristic when dealing with the inverses of linear functions. The slope and y-intercept of the inverse function are related to those of the original function, reflecting the swapped roles of input and output. Understanding how to find the inverse function is essential for solving equations, understanding function transformations, and various other mathematical applications.

c) Finding $f^{-1}(12)$: Evaluating the Inverse Function

Now that we have determined the inverse function, $f^{-1}(x) = 2x - 3$, we can evaluate it at specific points, just as we did with the original function. In this case, we are asked to find $f^{-1}(12)$, which means we need to substitute $x = 12$ into the expression for the inverse function.

Evaluating the inverse function at a specific point provides insight into the original function's behavior. Since the inverse function "undoes" the original function, evaluating $f^{-1}(12)$ tells us what input value to the original function would produce an output of 12. This connection between the function and its inverse is a key concept in understanding their relationship and applications.

To find $f^{-1}(12)$, we substitute $x = 12$ into the expression for $f^{-1}(x)$:

$f^{-1}(12) = 2(12) - 3$

Now, we perform the multiplication:

$f^{-1}(12) = 24 - 3$

Finally, we subtract:

$f^{-1}(12) = 21$

Therefore, $f^{-1}(12) = 21$. This result means that if we input 21 into the original function $f(x)$, the output will be 12. This confirms the inverse relationship between $f(x)$ and $f^{-1}(x)$.

This calculation demonstrates how the inverse function can be used to find the input value corresponding to a specific output value of the original function. The ability to evaluate inverse functions is crucial in various mathematical contexts, such as solving equations, understanding transformations, and working with composite functions. The result we obtained, $f^{-1}(12) = 21$, provides a concrete example of this inverse relationship, solidifying our understanding of the concept.

In this article, we have explored the concept of inverse functions using the example of a linear function, $f(x) = \frac{x+3}{2}$. We successfully found $f(2)$, determined the inverse function $f^{-1}(x) = 2x - 3$, and evaluated $f^{-1}(12)$. These steps provide a comprehensive understanding of how to work with functions and their inverses. Understanding the inverse of a function is a fundamental concept in mathematics, with applications spanning various fields. It allows us to reverse the operation of a function, providing valuable insights into the relationship between input and output values.

The process of finding an inverse function involves interchanging the roles of the input and output variables and then solving for the new output variable. This method is applicable to a wide range of functions, not just linear functions. The ability to determine and evaluate inverse functions is crucial for solving equations, understanding function transformations, and working with more complex mathematical concepts. The example we explored in this article serves as a solid foundation for further studies in mathematics and related disciplines.

By mastering the techniques demonstrated in this article, readers will be well-equipped to tackle problems involving functions and their inverses. The concepts discussed here are essential building blocks for more advanced mathematical topics, highlighting the importance of a thorough understanding of functions and their inverses. This exploration not only provides answers to the specific questions posed but also fosters a deeper appreciation for the elegance and interconnectedness of mathematical ideas.