Problem 52 - College Algebra with Modeling and Visualization

Question

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{2}{\sqrt{x}} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{4}{x^2} \).

Step 1: Identify the Domain of Original Function

First, ensure that the original function \( f(x) = \frac{2}{\sqrt{x}} \) is properly defined. The domain consists of all \( x > 0 \) because the square root function is not defined for non-positive \( x \) values, and \( x = 0 \) would cause division by zero.

Step 2: Swap and Solve for x

Interchange the roles of \( x \) and \( y \) in the original function to find its inverse. Rewrite \( y = \frac{2}{\sqrt{x}} \) as \( x = \frac{2}{\sqrt{y}} \). Now solve for \( y \).

Step 3: Square Both Sides

To eliminate the square root, square both sides of the equation \( x^2 = \left( \frac{2}{\sqrt{y}} \right)^2 \). This simplifies to \( x^2 = \frac{4}{y} \).

Step 4: Isolate y

Solve for \( y \) by multiplying both sides by \( y \) and then dividing both sides by \( x^2 \). We get \( y = \frac{4}{x^2} \).

Step 5: Conclude the Inverse Function

Since \( y \) represents \( f^{-1}(x) \), we have \( f^{-1}(x) = \frac{4}{x^2} \). Make sure this inverse function makes sense within the context of the domain of \( f(x) \), which implies \( x > 0 \).

Key Concepts

Domain of a function

Understanding the domain of a function is crucial in mathematics, especially when dealing with inverse functions. The domain of a function is the set of all possible input values (usually represented by the variable \( x \)) for which the function is defined. In simpler terms, it's the collection of \( x \) values where the function can operate without any mathematical errors.

For the function \( f(x) = \frac{2}{\sqrt{x}} \), the domain is restricted to \( x > 0 \). Why? Because for any square root, we cannot have a negative or zero under it in the real number system. This ensures that the expression under the square root is always positive, preventing undefined results. As division by zero is undefined, \( x = 0 \) would make the function invalid. In summary:

• Only positive numbers are allowed as the domain for \( x \).
• The domain excludes zero to avoid division by zero.

Square root functions

Square root functions are a special type of function that involve the square root of a variable. They take the form of \( \sqrt{x} \) and generate real outputs when \( x \) is non-negative. For example, \( \sqrt{4} = 2 \) because 2 squared results in 4. Therefore, these functions require an input greater than or equal to zero.

In the given function \( \frac{2}{\sqrt{x}} \), the square root is in the denominator, adding a level of complexity to the function. Not only does it need to be defined for positive \( x \), but it also influences the behavior and limitations of any inverse function calculations. Keep in mind:

• Negative inputs won't work as they aren't in the real number range for square roots.
• Zero as an input would cause division issues since \( \sqrt{0} \) is zero and division by zero is undefined.

Division by zero

Division by zero is a fundamental concept in mathematics that students need to grasp early on. It occurs when a number is divided by zero, which is undefined in the realm of real numbers. Why is this? Because dividing by zero does not yield a finite, meaningful result. For instance, if you try to divide 5 by 0, the result is undefined.

In the context of our exercise, the function \( \frac{2}{\sqrt{x}} \) needs careful attention to avoid this pitfall. When \( x = 0 \), the expression \( \frac{2}{\sqrt{x}} \) would imply a denominator of zero, hence leading to an undefined status. To avoid division by zero:

• Ensure that the domain excludes zero as an input.
• Always verify prior to calculations that none of the operations force a denominator to zero.

This is crucial to correctly identifying domains and inverse functions without stumbling into undefined territories.