Problem 149 - Algebra and Trigonometry

Question

Find the inverse of \(f(x)=x^{2}+4, x \geq 0\) (Section \(2.7, \text { Example } 7)\)

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = x^2 + 4\), for \(x \geq 0\), is \(f^{-1}(x) = \sqrt{x - 4}\).

Step 1: Swap Variables

First, replace \(f(x)\) with \(y\), to get \(y = x^2 + 4\). Now, swap \(x\) and \(y\) to get \(x = y^2 + 4\).

Step 2: Solve for y

Rearrange the equation to solve for \(y\), by subtracting 4 from both sides. This results in \(x - 4 = y^2\). Since we want to solve for \(y\), we must take the square root on both sides of the equation. Since \(x \geq 0\), we only take the positive root, giving us \(y = \sqrt{x - 4}\).

Step 3: Write the Inverse Function

Replace \(y\) with \(f^{-1}(x)\) to denote the inverse function. This gives us \(f^{-1}(x) = \sqrt{x - 4}\).

Key Concepts

Understanding Quadratic Functions

Quadratic functions are powerful tools in mathematics characterized by an equation of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. These functions are crucial as they form parabolas on the coordinate plane. The specific nature of a quadratic function’s graph is dictated by the coefficient \( a \). For instance, if \( a > 0 \), the parabola opens upwards; conversely, if \( a < 0 \), it opens downwards.

In our example, the quadratic function \( f(x) = x^2 + 4 \) represents a simple parabola with a vertical shift upwards by 4 units. This shift affects the graph's position but does not alter the parabola's shape, as there is no linear or constant term besides the shift. It’s important to note that the vertex of this parabola is at (0,4) and it only opens upward, making it continue infinitely in that direction. Understanding these basic properties helps in visualizing the graph and anticipating its behavior under transformations like finding inverses.

Domain Restrictions in Inverse Functions

Domain restrictions become essential when dealing with inverse functions, particularly for quadratic functions. To find an inverse, the original function must be one-to-one (bijective), which means it passes both horizontal and vertical line tests. However, since a full parabola fails the horizontal line test, domain restrictions ensure that the function is one-to-one.

In our problem, the domain restriction \( x \geq 0 \) is applied to ensure the quadratic function is invertible. This restriction effectively cuts the parabola in half, dealing only with the portion where \( x \) is non-negative. This part of the parabola allows us to establish a valid one-to-one correspondence between \( x \) and \( y \), where \( y \) is the function output. With such domain restrictions, we can definitively find the inverse function with coherent real values.

Solving Equations for Inverse Functions

Finding the inverse function involves a clear process of solving equations. First, swap the dependent and independent variables to express the function in terms of a new variable. In the example, this is represented by swapping \( x \) and \( y \) in the equation \( y = x^2 + 4 \), resulting in \( x = y^2 + 4 \).

Next, the task is to solve for the new variable, \( y \), to express it as a function of \( x \). Subtracting 4 from both sides gives \( x - 4 = y^2 \). The ability to solve for \( y \) is facilitated by taking the square root of both sides of the equation. Here, domain restrictions inform which root to consider. Since we have \( x \geq 0 \), only the positive square root is feasible, yielding \( y = \sqrt{x - 4} \). This step is crucial in articulating the inverse function, represented as \( f^{-1}(x) = \sqrt{x - 4} \). This systematic approach helps ensure that each transformation maintains mathematical validity, enabling students to find inverses confidently.