Exploring One-to-One Functions G And H And Their Inverses

In the captivating realm of mathematical functions, one-to-one functions stand out as unique entities, possessing the remarkable property of mapping each input to a distinct output. This article delves into the intricacies of two such functions, g and h, meticulously dissecting their definitions, inverses, and compositions. Prepare to embark on a journey of mathematical exploration as we unravel the secrets of these fascinating functions.

Unveiling the Essence of One-to-One Functions

At the heart of our exploration lies the concept of one-to-one functions, also known as injective functions. A function is deemed one-to-one if each element in its range corresponds to precisely one element in its domain. In simpler terms, no two distinct inputs can produce the same output. This distinctive characteristic sets one-to-one functions apart, allowing them to possess inverses – functions that effectively reverse the mapping, returning the original input for a given output.

Defining the One-to-One Function g

The function g is presented as a set of ordered pairs: g = {(-7, 8), (0, -4), (1, 3), (3, 1)}. Each ordered pair represents a mapping, where the first element is the input and the second element is the corresponding output. For instance, the pair (-7, 8) signifies that when the input is -7, the output of the function g is 8. Similarly, the pair (0, -4) indicates that an input of 0 yields an output of -4.

To ascertain whether g is indeed a one-to-one function, we meticulously examine its ordered pairs. We observe that each output value appears only once. The output 8 corresponds solely to the input -7, the output -4 is uniquely associated with the input 0, the output 3 maps exclusively to the input 1, and the output 1 is paired only with the input 3. This confirms that no two distinct inputs produce the same output, thereby establishing g as a one-to-one function. The function g's one-to-one nature is crucial for the existence of its inverse, which we will explore later.

Defining the One-to-One Function h

In contrast to g, the function h is defined by an algebraic expression: h(x) = 3x + 14. This expression provides a rule for calculating the output for any given input x. For instance, if we input x = 2, the output of h would be h(2) = 3(2) + 14 = 20. Similarly, for x = -1, we get h(-1) = 3(-1) + 14 = 11. To verify that h is a one-to-one function, we can employ the horizontal line test. Imagine drawing a horizontal line across the graph of h(x) = 3x + 14. If this line intersects the graph at most once, then the function is one-to-one. Since h(x) is a linear function with a non-zero slope (3 in this case), any horizontal line will intersect it at exactly one point, confirming that h is indeed a one-to-one function. This one-to-one property of h(x) = 3x + 14 makes it invertible, allowing us to find its inverse function.

Finding the Inverse of g: Unraveling the Reverse Mapping

Now that we've established g as a one-to-one function, we can embark on the quest to find its inverse, denoted as g⁻¹(x). The inverse function essentially reverses the mapping of g. If g maps an input x to an output y, then g⁻¹ maps y back to x. To find g⁻¹(3), we seek the input value in the original function g that produces an output of 3. Examining the ordered pairs of g, we observe that the pair (1, 3) satisfies this condition. This signifies that when the input is 1, the output of g is 3. Therefore, the inverse function g⁻¹ maps 3 back to 1, and we can confidently state that g⁻¹(3) = 1. Finding g⁻¹(3) involves tracing back from the output 3 to its corresponding input in the original function g.

Determining the Inverse of h: Reversing the Algebraic Expression

Finding the inverse of h, denoted as h⁻¹(x), requires a slightly different approach, as h is defined by an algebraic expression. Our goal is to find an expression that reverses the operation performed by h. Recall that h(x) = 3x + 14. To reverse this, we first undo the addition of 14 by subtracting 14 from both sides, resulting in h(x) - 14 = 3x. Next, we undo the multiplication by 3 by dividing both sides by 3, yielding (h(x) - 14) / 3 = x. Finally, we replace h(x) with x and x with h⁻¹(x) to obtain the inverse function: h⁻¹(x) = (x - 14) / 3. This expression now represents the inverse function h⁻¹, which takes an output x and returns the corresponding input. The process of finding h⁻¹(x) involves reversing the operations performed by the original function h(x).

Unveiling the Composition of h⁻¹ and h: A Journey Back to the Origin

The final part of our exploration involves the composition of h⁻¹ and h, denoted as (h⁻¹ ∘ h)(x). Function composition essentially means applying one function to the result of another. In this case, we first apply h to an input x, obtaining h(x), and then apply h⁻¹ to the result, h⁻¹(h(x)). A crucial property of inverse functions is that their composition results in the original input. In other words, (h⁻¹ ∘ h)(x) = x for any input x. To find (h⁻¹ ∘ h)(-5), we can directly apply this property. Since h⁻¹ and h are inverses of each other, their composition will simply return the original input, which is -5. Therefore, (h⁻¹ ∘ h)(-5) = -5. This demonstrates the fundamental relationship between a function and its inverse, where their composition effectively cancels each other out, leaving the initial input unchanged. The composition (h⁻¹ ∘ h)(-5) illustrates the property of inverse functions where they undo each other's operations.

Conclusion: A Symphony of Mathematical Concepts

Our exploration of the one-to-one functions g and h has unveiled a symphony of mathematical concepts, from the fundamental definition of one-to-one functions to the intricacies of finding inverses and composing functions. We've witnessed how the unique property of one-to-one functions allows them to possess inverses, and how these inverses effectively reverse the mapping of the original functions. The composition of a function and its inverse has demonstrated a beautiful symmetry, returning us to the initial input. This journey into the realm of one-to-one functions has not only provided concrete answers to the posed questions but also illuminated the profound connections and elegance within the world of mathematics. Understanding one-to-one functions is crucial for further studies in advanced mathematical concepts.