Problem 42 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=x^{2}+x, \quad x \geq-\frac{1}{2}\)

Step-by-Step Solution

Answer

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

Step 1: Replace f(x) with y

Begin by replacing the function notation \( f(x) = x^2 + x \) with \( y = x^2 + x \). This helps set up the equation for finding the inverse.

Step 2: Swap variables

To find the inverse, swap \(x\) and \(y\) in the equation. The equation becomes \( x = y^2 + y \). This represents the equation of the inverse function.

Step 3: Rearrange the equation for y

Rearrange the equation \( x = y^2 + y \) to isolate \(y\). Start by completing the square on the right side to facilitate solving for \(y\).

Step 4: Complete the square

For \( y^2 + y \), add and subtract \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \) inside the equation. This turns it into \((y + \frac{1}{2})^2 - \frac{1}{4}\). Rewrite the equation as \( x + \frac{1}{4} = (y + \frac{1}{2})^2 \).

Step 5: Solve for y

Take the square root of both sides: \(y + \frac{1}{2} = \pm \sqrt{x + \frac{1}{4}}\). Since \(x \geq -\frac{1}{2} \), use the positive root to ensure the function is one-to-one. This gives \( y = \sqrt{x + \frac{1}{4}} - \frac{1}{2} \).

Step 6: Write the inverse function

The inverse function is \( f^{-1}(x) = \sqrt{x + \frac{1}{4}} - \frac{1}{2} \), valid for \( x \geq -\frac{1}{4} \). Check that the domain and range are consistent with the original function.

Key Concepts

Completing the Square

Completing the square is a technique used to solve quadratic equations and is particularly useful when finding the inverse of certain functions. The idea is to transform a quadratic equation from the standard form, \( ax^2 + bx + c \), into a perfect square plus a constant. This makes it easier to solve or rearrange, especially when dealing with variables that need to be isolated.

For instance, consider the quadratic expression \( y^2 + y \). We can "complete the square" by making it look like \((y + d)^2 - e\). To achieve this, focus on the coefficient of \( y \), which is 1 in our example. Take half of this value (\( \frac{1}{2}\)) and square it (\( \frac{1}{2}^2 = \frac{1}{4} \)). Add and subtract this squared term within the expression:

• Add \( \frac{1}{4} \) to create a perfect square: \( (y + \frac{1}{2})^2 \)
• Subtract \( \frac{1}{4} \) to balance the equation.

Completing the square helps create a straightforward equation that can be manipulated to isolate variables and solve for them.

Function Notation

Function notation is a way of expressing relationships between variables. In mathematics, you'll often see expressions written as \( f(x) \). This notation conveys that \( f \) is a function where \( x \) is the input, and \( f(x) \) (or \( y \)) is the output. It's a compact method for describing functions and understanding how changes in the input affect the output.

In our example, we started with \( f(x) = x^2 + x \). By using function notation, mathematicians can easily swap out values or even variables to explore different scenarios, such as finding inverse functions. When solving for the inverse, the notation \( f^{-1}(x) \) is used to denote that you're finding a function that "undoes" \( f \). Thus, if \( y = f(x) \), then \( x = f^{-1}(y) \). This notation is essential for clearly indicating which function or form you’re working with at any stage of the problem.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. These equations are typically in the form \( ax^2 + bx + c = 0 \) and can be solved using various methods. One of the most flexible methods is using the quadratic formula, but another method is "completing the square," as discussed earlier.

When you're dealing with quadratics in the context of inverse functions, solving them often involves finding a way to isolate the variable of interest, such as \( y \) in the equation \( x = y^2 + y \). After rearranging and completing the square, we obtained \( (y + \frac{1}{2})^2 = x + \frac{1}{4} \). Solving the square yields two potential solutions, reflected as \( y + \frac{1}{2} = \pm \sqrt{x + \frac{1}{4}} \). However, context dictates which solution to choose. In our case, ensuring the function is one-to-one leads us to choose only the positive root.

• Begin with the quadratic set to zero or isolated form.
• Use "completing the square" or apply the quadratic formula.
• Choose the appropriate solution based on the constraints of the problem.

Through these steps, you can effectively solve equations that arise when finding inverses or resolving various algebraic problems.