Inverse Of F(x) = X^2 + 3: Domain And Solution

Let's dive into the world of inverse functions with a specific example: f(x) = x² + 3, where the domain is restricted to [0, ∞). In this comprehensive guide, we'll walk through the process of finding the inverse function, denoted as f⁻¹(x), and determining its domain. Understanding inverse functions is a crucial concept in mathematics, especially in areas like calculus and real analysis. This article aims to provide a clear and detailed explanation, making it accessible for learners of all levels.

Understanding Inverse Functions

Before we jump into the specifics of our function, let's clarify what an inverse function actually is. Think of a function as a machine: you put something in (an input, x), and it spits something else out (an output, f(x)). An inverse function, f⁻¹(x), is like that machine in reverse. You put in what the original function spat out, and it gives you back the original input. More formally, if f(a) = b, then f⁻¹(b) = a. This fundamental property is what allows us to "undo" the operation of the original function. However, not every function has an inverse. For a function to have an inverse, it must be one-to-one, also known as injective. This means that each output value corresponds to exactly one input value. Graphically, a one-to-one function passes the horizontal line test: no horizontal line intersects the graph more than once. The function f(x) = x² + 3 restricted to the domain [0, ∞) is indeed one-to-one, as we'll see later when we analyze its graph.

Why is the Domain Restriction Important?

You might wonder why we specifically mentioned the domain [0, ∞) for f(x) = x² + 3. Without this restriction, the function would not be one-to-one. Consider the full quadratic function without any domain restrictions. Both positive and negative x values would yield the same y value when squared. For example, both 2 and -2, when plugged into x², yield 4. This violates the one-to-one requirement for a function to have an inverse. By restricting the domain to non-negative numbers [0, ∞), we ensure that each output has only one corresponding input. This restriction is crucial for the existence of the inverse function. The restricted domain effectively cuts the parabola in half, taking only the right side where the function is increasing and hence one-to-one. This concept of restricting the domain to ensure a function has an inverse is a common technique in mathematics, especially when dealing with functions that are not inherently one-to-one over their entire natural domain. We'll see this concept again when we discuss trigonometric functions and their inverses.

Finding the Inverse Function: Step-by-Step

Now that we understand the importance of one-to-one functions and domain restrictions, let's get to the heart of the matter: finding the inverse of f(x) = x² + 3. We'll follow a systematic approach to ensure clarity and accuracy.

Step 1: Replace f(x) with y

This is a simple notational change to make the algebra cleaner. We rewrite the function as:

y = x² + 3

Step 2: Swap x and y

This is the core step in finding the inverse. We're essentially reversing the roles of input and output. This gives us:

x = y² + 3

Step 3: Solve for y

Our goal now is to isolate y on one side of the equation. This will give us the equation for the inverse function. First, subtract 3 from both sides:

x - 3 = y²

Next, take the square root of both sides. Remember that taking the square root introduces both positive and negative solutions. However, since our original function had a domain restriction of [0, ∞), we know that the inverse function's range will also be [0, ∞). Therefore, we only consider the positive square root:

y = √(x - 3)

Step 4: Replace y with f⁻¹(x)

This is the final step in expressing our result in the correct notation. We replace y with f⁻¹(x) to denote the inverse function:

f⁻¹(x) = √(x - 3)

And there you have it! We've successfully found the inverse function of f(x) = x² + 3, which is f⁻¹(x) = √(x - 3).

Determining the Domain of the Inverse Function

Finding the inverse function is only half the battle. We also need to determine its domain. The domain of the inverse function is closely related to the range of the original function. In fact, the domain of f⁻¹(x) is equal to the range of f(x). This makes intuitive sense, as the inputs of the inverse function are the outputs of the original function.

Finding the Range of f(x)

To find the range of f(x) = x² + 3 on the domain [0, ∞), we need to consider the possible output values. Since x² is always non-negative for any real number x, and we're adding 3, the smallest possible value of f(x) occurs when x = 0. In this case, f(0) = 0² + 3 = 3. As x increases, x² also increases, so f(x) will continue to increase without bound. Therefore, the range of f(x) is [3, ∞).

The Domain of f⁻¹(x)

Since the domain of f⁻¹(x) is equal to the range of f(x), the domain of f⁻¹(x) = √(x - 3) is [3, ∞). We can also see this directly from the inverse function itself. The square root function is only defined for non-negative values. Therefore, x - 3 must be greater than or equal to 0, which means x ≥ 3. This confirms our result that the domain of f⁻¹(x) is [3, ∞).

Summary and Key Takeaways

Let's recap what we've learned in this article:

• We started with the function f(x) = x² + 3 and restricted its domain to [0, ∞) to ensure it was one-to-one.
• We walked through the steps of finding the inverse function, f⁻¹(x), by swapping x and y and solving for y.
• We found that f⁻¹(x) = √(x - 3).
• We determined the range of f(x) to be [3, ∞).
• We concluded that the domain of f⁻¹(x) is [3, ∞), which is equal to the range of f(x).

Understanding inverse functions is a critical skill in mathematics. By mastering the techniques outlined in this article, you'll be well-equipped to tackle more complex problems involving inverse functions and their properties. Remember to always check the domain and range when working with inverse functions, as they are closely intertwined. By practice and patience, finding inverse functions will become second nature.

For further exploration and a deeper understanding of inverse functions, you might find the resources available at Khan Academy particularly helpful. They offer a variety of explanations, examples, and practice problems to solidify your knowledge.