One-to-One Functions And Inverses: Solving For G⁻¹(3), H⁻¹(x), And (h⁻¹ ∘ H)(-5)

This article delves into the concept of one-to-one functions, focusing on how to determine their inverses and compositions. We'll explore these ideas through the lens of two specific functions: a discrete function g defined by ordered pairs and a linear function h. Understanding these functions is crucial for mastering fundamental concepts in mathematics. Let's embark on a journey to understand one-to-one functions, their inverses, and compositions.

Understanding One-to-One Functions

A one-to-one function, also known as an injective function, is a function where each element in the range corresponds to exactly one element in the domain. In simpler terms, no two different inputs produce the same output. This unique characteristic allows one-to-one functions to have inverses. Determining if a function is one-to-one is crucial before attempting to find its inverse. Several methods can be employed to ascertain if a function possesses this property, including the horizontal line test for graphically represented functions and algebraic verification for functions defined by equations or sets of ordered pairs. In this article, we will primarily focus on finding the inverses and compositions of the given one-to-one functions g and h, and this foundational understanding is essential for more advanced mathematical concepts.

Function g and Finding g⁻¹(3)

The function g is defined as a set of ordered pairs: g = {(-7, 8), (0, -4), (1, 3), (3, 1)}. To find g⁻¹(3), we need to understand what an inverse function represents. The inverse function, denoted as g⁻¹, essentially reverses the mapping of the original function g. If g(x) = y, then g⁻¹(y) = x. In other words, the inverse function takes an output of the original function and returns the corresponding input. When dealing with ordered pairs, finding the inverse is straightforward: simply swap the x and y values in each pair. Therefore, to find g⁻¹(3), we look for the ordered pair in g where the y-value is 3. We observe the ordered pair (1,3) in g. This means that g(1) = 3, and consequently, g⁻¹(3) = 1. Therefore, the value of g⁻¹(3) is 1. This example demonstrates the fundamental principle of inverse functions: reversing the input-output relationship of the original function. The concept of inverse functions is paramount in various mathematical domains, including solving equations, understanding transformations, and exploring mathematical relationships. Mastering the ability to find and interpret inverse functions is a key step in developing a strong mathematical foundation.

Function h and Finding h⁻¹(x)

The function h is defined by the equation h(x) = 3x + 14. This is a linear function, and since it's a linear function with a non-zero slope, it's guaranteed to be one-to-one. To find the inverse function h⁻¹(x), we need to reverse the operations performed by h. The function h first multiplies the input x by 3 and then adds 14. To reverse these operations, we'll first subtract 14 and then divide by 3. Let y = h(x) = 3x + 14. To find the inverse, we solve for x in terms of y:

y = 3x + 14

Subtract 14 from both sides:

y - 14 = 3x

Divide both sides by 3:

x = (y - 14) / 3

Now, we swap x and y to express the inverse function in the standard form h⁻¹(x): h⁻¹(x) = (x - 14) / 3. This equation represents the inverse function of h. For any input x, h⁻¹(x) will return the value that, when input into h, would produce x. Finding the inverse of a function is a fundamental skill in mathematics, allowing us to solve equations, analyze relationships between variables, and understand the reverse process of a given function. The ability to manipulate equations and isolate variables is crucial for determining inverse functions, and this process highlights the interconnectedness of mathematical operations.

Composition of Functions and Finding (h⁻¹ ∘ h)(-5)

The notation (h⁻¹ ∘ h)(-5) represents the composition of functions. Specifically, it means we first evaluate h at -5, and then we take that result and evaluate h⁻¹ at that value. In general, ( f ∘ g )(x) = f(g(x)). In simpler terms, we apply the function g to x first, and then we apply the function f to the result. In the case of (h⁻¹ ∘ h)(-5), we start by finding h(-5). We know that h(x) = 3x + 14, so h(-5) = 3*(-5) + 14 = -15 + 14 = -1. Now, we need to find h⁻¹(-1). We previously found that h⁻¹(x) = (x - 14) / 3, so h⁻¹(-1) = (-1 - 14) / 3 = -15 / 3 = -5. Therefore, (h⁻¹ ∘ h)(-5) = -5. This result is not coincidental. The composition of a function and its inverse always results in the original input. In other words, (h⁻¹ ∘ h)(x) = x for all x in the domain of h. This property underscores the fundamental relationship between a function and its inverse, highlighting how they