Finding The Inverse Of F(x) = X² + 3 A Comprehensive Guide

In mathematics, the concept of an inverse function is fundamental. It's like having a mathematical undo button. If a function f takes an input x and produces an output y, the inverse function, denoted as f⁻¹, takes that output y and returns the original input x. However, not all functions have inverses. For a function to have an inverse, it must be one-to-one, meaning that each output corresponds to only one input. This article delves into the function f(x) = x² + 3, where x ≥ 0, a classic example used to illustrate inverse functions. We will explore its one-to-one nature, find its inverse, determine the domain and range of both f and f⁻¹, and visualize their relationship graphically.

Verifying the One-to-One Property

The function f(x) = x² + 3 is defined for x ≥ 0. This restriction is crucial because the unrestricted quadratic function x² + 3 is not one-to-one due to the symmetry of the parabola. However, by limiting the domain to non-negative values of x, we effectively consider only the right half of the parabola, which is indeed one-to-one. To rigorously prove this, we can use the horizontal line test. Imagine drawing a horizontal line across the graph of f(x) = x² + 3 for x ≥ 0. This line will intersect the graph at most once, confirming that the function is one-to-one within the specified domain. Another way to verify the one-to-one property is through an algebraic approach. Suppose f(x₁) = f(x₂). This means x₁² + 3 = x₂² + 3. Subtracting 3 from both sides gives x₁² = x₂². Taking the square root of both sides yields |x₁| = |x₂|. Since both x₁ and x₂ are greater than or equal to zero (due to the domain restriction), this implies that x₁ = x₂. Thus, we have mathematically demonstrated that if the function values are equal, the inputs must also be equal, which is the defining characteristic of a one-to-one function.

Finding the Inverse Function

Now that we've established that f(x) = x² + 3 for x ≥ 0 is one-to-one, we can proceed to find its inverse. The process involves a few key steps. First, we replace f(x) with y: y = x² + 3. Next, we swap x and y: x = y² + 3. This is the crucial step in finding the inverse, as we are essentially reversing the roles of input and output. Now, we solve for y. Subtracting 3 from both sides gives x - 3 = y². Taking the square root of both sides yields y = ±√(x - 3). Since the range of the inverse function corresponds to the domain of the original function (which is x ≥ 0), we choose the positive square root. Therefore, the inverse function is f⁻¹(x) = √(x - 3). To check our answer, we can verify that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Let's compute f(f⁻¹(x)): f(√(x - 3)) = (√(x - 3))² + 3 = (x - 3) + 3 = x. Similarly, let's compute f⁻¹(f(x)): f⁻¹(x² + 3) = √((x² + 3) - 3) = √(x²) = |x|. Since x ≥ 0, we have |x| = x. Both compositions result in x, confirming that f⁻¹(x) = √(x - 3) is indeed the inverse of f(x) = x² + 3 for x ≥ 0.

Determining the 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 function f(x) = x² + 3 with x ≥ 0, the domain is explicitly given as [0, ∞), which includes all non-negative real numbers. To find the range, we consider the behavior of the function. As x varies from 0 to infinity, x² also varies from 0 to infinity. Adding 3 to x² shifts the entire range upwards by 3 units. Therefore, the range of f(x) is [3, ∞), which includes all real numbers greater than or equal to 3. For the inverse function f⁻¹(x) = √(x - 3), the domain is determined by the requirement that the expression inside the square root must be non-negative. Thus, x - 3 ≥ 0, which implies x ≥ 3. So, the domain of f⁻¹(x) is [3, ∞). The range of f⁻¹(x) corresponds to the domain of the original function f(x), which is [0, ∞). This is because the inverse function essentially reverses the roles of input and output. The domain of f becomes the range of f⁻¹, and vice versa. Understanding the interplay between the domains and ranges of a function and its inverse is crucial for a comprehensive understanding of their relationship.

Visualizing the Functions Graphically

Graphing functions provides a visual representation of their behavior and relationships. Let's consider the graphs of f(x) = x² + 3, f⁻¹(x) = √(x - 3), and the line y = x on the same coordinate axes. The graph of f(x) = x² + 3 for x ≥ 0 is the right half of a parabola that opens upwards, with its vertex at the point (0, 3). The graph of f⁻¹(x) = √(x - 3) is the upper half of a parabola that opens to the right, with its starting point at (3, 0). The line y = x is a straight line that passes through the origin with a slope of 1. A key observation is that the graphs of f(x) and f⁻¹(x) are reflections of each other across the line y = x. This is a fundamental property of inverse functions. The line y = x acts as a mirror, and the reflection of the graph of f(x) across this line produces the graph of f⁻¹(x), and vice versa. This visual symmetry underscores the idea that inverse functions essentially