Finding The Inverse Function Of F(x) = (4x - 5) / (x + 5)

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In the realm of mathematics, inverse functions play a pivotal role in unraveling the relationships between variables and their corresponding transformations. When confronted with a function, a natural question arises: can we reverse its operation? In other words, can we find a function that undoes the effects of the original function? This quest leads us to the concept of inverse functions, which are essential tools for solving equations, understanding functional relationships, and exploring mathematical structures.

In this article, we embark on a journey to determine the inverse function, denoted as f⁻¹(x), of the given function f(x) = (4x - 5) / (x + 5). This exploration will not only unveil the mechanics of finding inverse functions but also highlight the underlying principles and their significance in mathematical analysis. To begin, let's first grasp the fundamental concept of inverse functions.

Grasping the Essence of Inverse Functions

At its core, an inverse function acts as a mirror image of the original function. If a function f(x) maps an input x to an output y, then its inverse function f⁻¹(x) reverses this mapping, taking the output y back to the original input x. Mathematically, this relationship is expressed as follows:

  • f⁻¹(f(x)) = x for all x in the domain of f
  • f(f⁻¹(x)) = x for all x in the range of f

This symmetry between a function and its inverse is a defining characteristic. Graphically, the graphs of f(x) and f⁻¹(x) are reflections of each other across the line y = x. This visual representation provides a powerful tool for understanding the relationship between a function and its inverse.

However, not all functions possess an inverse. For a function to have an inverse, it must be one-to-one, also known as injective. This means that each input value maps to a unique output value. In other words, no two different inputs can produce the same output. This one-to-one property ensures that the inverse function can unambiguously map an output back to its original input.

Unveiling the Steps to Find the Inverse Function

Now that we have a solid understanding of inverse functions, let's delve into the process of finding the inverse of f(x) = (4x - 5) / (x + 5). The general procedure involves the following steps:

  1. Replace f(x) with y: This substitution simplifies the notation and prepares the equation for the subsequent steps. In our case, we have:

    y = (4x - 5) / (x + 5)

  2. Swap x and y: This step embodies the essence of finding the inverse, as we are reversing the roles of input and output. Swapping x and y in the equation, we get:

    x = (4y - 5) / (y + 5)

  3. Solve for y: This is the crucial step where we isolate y on one side of the equation. To do this, we first multiply both sides by (y + 5):

    x(y + 5) = 4y - 5

    Expanding the left side, we have:

    xy + 5x = 4y - 5

    Now, we want to group the terms containing y on one side and the other terms on the other side. Subtracting 4y from both sides and subtracting 5x from both sides, we get:

    xy - 4y = -5x - 5

    Next, we factor out y from the left side:

    y(x - 4) = -5x - 5

    Finally, we divide both sides by (x - 4) to isolate y:

    y = (-5x - 5) / (x - 4)

  4. Replace y with f⁻¹(x): This final step expresses the inverse function using the standard notation:

    f⁻¹(x) = (-5x - 5) / (x - 4)

Thus, we have successfully found the inverse function of f(x) = (4x - 5) / (x + 5), which is f⁻¹(x) = (-5x - 5) / (x - 4).

Delving Deeper: Domain and Range Considerations

Having found the inverse function, it's crucial to consider its domain and range. The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

For the original function f(x) = (4x - 5) / (x + 5), the domain is all real numbers except x = -5, as this value would make the denominator zero, leading to an undefined expression. The range of f(x) is all real numbers except y = 4. This can be seen by considering the horizontal asymptote of the function.

Now, for the inverse function f⁻¹(x) = (-5x - 5) / (x - 4), the domain is all real numbers except x = 4, and the range is all real numbers except y = -5. Notice that the domain of f⁻¹(x) is the range of f(x), and the range of f⁻¹(x) is the domain of f(x). This reciprocal relationship between the domains and ranges of a function and its inverse is a general property of inverse functions.

Verifying the Inverse Function: A Crucial Step

To ensure that we have indeed found the correct inverse function, we need to verify that it satisfies the defining properties of inverse functions:

  • f⁻¹(f(x)) = x for all x in the domain of f
  • f(f⁻¹(x)) = x for all x in the range of f

Let's verify the first property:

f⁻¹(f(x)) = f⁻¹((4x - 5) / (x + 5))

Substituting (4x - 5) / (x + 5) into f⁻¹(x), we get:

f⁻¹(f(x)) = (-5((4x - 5) / (x + 5)) - 5) / (((4x - 5) / (x + 5)) - 4)

To simplify this expression, we multiply the numerator and denominator by (x + 5):

f⁻¹(f(x)) = (-5(4x - 5) - 5(x + 5)) / ((4x - 5) - 4(x + 5))

Expanding and simplifying, we have:

f⁻¹(f(x)) = (-20x + 25 - 5x - 25) / (4x - 5 - 4x - 20)

f⁻¹(f(x)) = (-25x) / (-25) = x

Thus, the first property is satisfied.

Now, let's verify the second property:

f(f⁻¹(x)) = f((-5x - 5) / (x - 4))

Substituting (-5x - 5) / (x - 4) into f(x), we get:

f(f⁻¹(x)) = (4((-5x - 5) / (x - 4)) - 5) / (((-5x - 5) / (x - 4)) + 5)

Multiplying the numerator and denominator by (x - 4), we have:

f(f⁻¹(x)) = (4(-5x - 5) - 5(x - 4)) / ((-5x - 5) + 5(x - 4))

Expanding and simplifying, we get:

f(f⁻¹(x)) = (-20x - 20 - 5x + 20) / (-5x - 5 + 5x - 20)

f(f⁻¹(x)) = (-25x) / (-25) = x

The second property is also satisfied. Therefore, we have verified that f⁻¹(x) = (-5x - 5) / (x - 4) is indeed the inverse function of f(x) = (4x - 5) / (x + 5).

Real-World Applications and Significance

Inverse functions are not merely abstract mathematical concepts; they have numerous applications in various fields, including:

  • Cryptography: Inverse functions are used to encrypt and decrypt messages, ensuring secure communication.
  • Computer Graphics: Inverse transformations are used to manipulate and display objects in three-dimensional space.
  • Calculus: Inverse functions are essential for finding antiderivatives and solving differential equations.
  • Economics: Inverse functions are used to model supply and demand curves and analyze market equilibrium.

Beyond these specific applications, inverse functions provide a fundamental understanding of how mathematical operations can be reversed. This concept is crucial for problem-solving, mathematical reasoning, and a deeper appreciation of the interconnectedness of mathematical ideas.

Conclusion: A Journey into the Realm of Inverse Functions

In this article, we have embarked on a comprehensive exploration of inverse functions, focusing on the specific example of f(x) = (4x - 5) / (x + 5). We have not only unveiled the steps to find the inverse function, f⁻¹(x) = (-5x - 5) / (x - 4), but also delved into the underlying concepts, domain and range considerations, verification procedures, and real-world applications. Understanding inverse functions is not just about mastering a mathematical technique; it's about developing a deeper understanding of mathematical relationships and their significance in the world around us. So, embrace the power of inverse functions and unlock new mathematical insights!