Problem 71 - Algebra 2

Question

What is the inverse of \(y=x^{2}-3 ?\) $$ \begin{array}{ll}{\text { A. } y=\pm \sqrt{x}+3} & {\text { B. } y=\pm \sqrt{x}-3} \\ {\text { C. } y=\pm \sqrt{x+3}} & {\text { D. } y=\pm \sqrt{x-3}}\end{array} $$

Step-by-Step Solution

Answer

The inverse of \(y=x^{2}-3\) is \(y = \pm \sqrt{x + 3}\). So, the correct answer is option C.

Step 1: Swap y and x

Start by swapping \(y\) and \(x\) in the original equation. This will give \(x = y^{2} - 3\).

Step 2: Solve for y

To find the inverse, solve the equation from Step 1 for \(y\). First isolate \(y^{2}\) by adding 3 to both sides, resulting in \(x + 3 = y^{2}\). Then, to isolate \(y\), find the square root of both sides. This gives \(y = \pm \sqrt{x + 3}\).

Step 3: Check against possible answers

By comparing the result from Step 2 to the given options, it's clear that the correct answer matches with option C. Hence, the inverse of \(y=x^{2}-3\) is \(y = \pm \sqrt{x + 3}\).

Key Concepts

Quadratic Equations

Quadratic equations are equations that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations are called "quadratic" because the highest exponent on the variable \(x\) is 2, giving them a characteristic parabolic graph shape. When solving quadratic equations, our goal is to determine the values of \(x\) that make the equation true. These solutions are called "roots."

• The roots can be real or complex numbers. If they are real, they are the x-intercepts of the parabola formed by the quadratic equation.
• Quadratic equations can be solved using various methods such as factoring, using the quadratic formula, completing the square, or graphically.

For the exercise, the expression \(y = x^2 - 3\) represents a specific type of a quadratic function. When an inverse function is involved, the quadratic must be carefully managed, often by swapping variables to solve for the original input in terms of the output. This leads us naturally to the concept of a square root in the next section.

Square Root

The square root of a number or expression \(x\) is a value that, when multiplied by itself, yields \(x\). It is often denoted as \(\sqrt{x}\). In the context of solving equations or finding inverse functions, square roots play a crucial role, especially when dealing with quadratic expressions.

• The square root function is the inverse operation of squaring. Therefore, when you take the square root of both sides of an equation, you're essentially reversing the process of squaring.
• However, solving for a square root introduces two potential solutions: the positive and negative roots, as \(y = \pm \sqrt{x}\). This is because both positive and negative values, when squared, produce the same result.

In our exercise, to isolate \(y\), we took the square root of both sides. This solution resulted in \(y = \pm \sqrt{x + 3}\). This step is critical for understanding how inverse functions work when dealing with quadratics.

Solving Equations

Solving equations involves finding the value(s) of the variable that make the equation true. This process may vary depending on the type of equation, like linear or quadratic. For quadratics, one often considers solving by isolating terms and using inverse operations, like the square root, to undo operations.

• In the example problem, solving the equation involves isolating the \(y^2\) term by altering the arrangement of the original equation, and then applying the square root to derive the inverse function.
• Every step has to be followed with caution, especially when performing operations like taking a square root, which introduces both positive and negative solutions.

The inverse of a function exchanges the roles of inputs and outputs. So in solving for \(y\) in \(x = y^2 - 3\), the steps required us to carefully apply mathematical operations to express \(y\) solely in terms of \(x\), yielding the solution \(y = \pm \sqrt{x+3}\). This method not only gives you the inverse but also reinforces the intuitive understanding of solving and transforming equations.