Problem 72 - Algebra 2

Question

What is the inverse of \(y=4 x^{2}+5 ?\) For what values of \(x\) is the inverse a real number?

Step-by-Step Solution

Answer

The inverse of \(y=4 x^{2}+5\) is \(y = \sqrt{(x-5)/4}\). The inverse is a real number for \(x\) values greater than or equal to 5.

Step 1: Write out the original function

The original function is \(y=4 x^{2}+5\).

Step 2: Switch \(x\) and \(y\)

This results in the equation \(x=4 y^{2}+5\).

Step 3: Solve the equation for \(y\)

To do this, first subtract 5 from both sides which results in \(x - 5 = 4 y^{2}\). Then divide both sides by 4, which gives \((x-5)/4 = y^{2}\).

Step 4: Take the square root of both sides

This results in \(y = \pm \sqrt{(x-5)/4}\). However, since \( y^{2} = (x-5)/4 \), the original function \(y\) was a parabola that opens upwards, the inverse will only include the positive square root as it has to be a function. Therefore, the inverse is \(y = \sqrt{(x-5)/4}\).

Step 5: Determine for what values of \(x\) the inverse is a real number

For \(y = \sqrt{(x-5)/4}\) to be real, the expression under the square root \((x-5)/4\) must be greater than or equal to zero, as you cannot take the square root of a negative number. This implies \(x-5 \geq 0\) or \(x \geq 5\). So for \(x\) values greater than or equal to 5, the inverse is a real number.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function represented by an equation of the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In the given problem, the quadratic function is \(y = 4x^2 + 5\). Notice how this function contains the \(x^2\) term, which gives it a characteristic parabolic shape on a graph. Quadratic functions have several interesting properties:

• They are always symmetrical about a vertical line called the axis of symmetry.
• The highest or lowest point on the graph is called the vertex.
• The parabola opens upwards if \(a>0\) and downwards if \(a<0\).

It's essential to understand these features because they help determine the inverse's behavior. When finding the inverse of a quadratic function, ensure to consider these properties and how they affect the resulting function's range and domain.

Solving Equations

Solving equations is a fundamental skill required for finding the inverse of a quadratic function. The process involves manipulating the equation to isolate the desired variable. In our problem, after switching \(x\) and \(y\) in the original equation, we get \(x = 4y^2 + 5\). To find \(y\), perform the following steps:

• Subtract 5 from each side: \(x - 5 = 4y^2\).
• Divide both sides by 4: \(\frac{x-5}{4} = y^2\).

Each step systematically transforms the equation, bringing \(y\) closer to being isolated. It's important to use these algebraic techniques in sequence to correctly solve for \(y\). Avoid skipping steps to maintain accuracy in calculations.

Domain of a Function

The domain of a function refers to the set of all possible input values (usually \(x\)) for which the function is defined. For the inverse function \(y = \sqrt{(x-5)/4}\), determining the domain involves ensuring that the expression inside the square root remains non-negative. This requirement arises because a square root operation on a negative number results in an imaginary number, which isn't included in the real number system. Thus, we solve the inequality:

• \(\frac{x-5}{4} \geq 0\)
• This simplify to \(x - 5 \geq 0\), giving \(x \geq 5\).

Hence, the domain of the inverse function is all real numbers \(x\) such that \(x \geq 5\). This analysis ensures that the function outputs real numbers, making it a viable function in real-world scenarios.

Square Root Operation

The square root operation is an essential part of finding the inverse of a quadratic function. It involves determining a value that, when multiplied by itself, gives the original number. For our problem, after solving \(\frac{x-5}{4} = y^2\), taking the square root of both sides gives \(y = \pm \sqrt{\frac{x-5}{4}}\). Here, the "±" indicates that both a positive and a negative solution exist.However, for the inverse to remain a function (a relation where each input corresponds to a single output), we select only the principal square root (the positive value). This choice retains the function's integrity and adheres to the definition of functions we use in mathematics. Therefore, \(y = \sqrt{\frac{x-5}{4}}\) is the meaningful inverse from a functional perspective, mapping each input to one clear output.