Problem 42 - Algebra and Trigonometry

Question

Find the inverse function of \(f\) $$ f(x)=\frac{1}{x^{2}}, \quad x>0 $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{1}{\sqrt{x}}\), for \(x > 0\).

Step 1: Understand the function

The function given is \[f(x) = \frac{1}{x^2} \quad \text{for} \; x > 0\]We are being asked to find the inverse of this function.

Step 2: Write the equation for the inverse

To find the inverse function, we start by setting \[y = \frac{1}{x^2}\]Next, we will solve for \(x\) in terms of \(y\).

Step 3: Solve for x

Multiply both sides by \(x^2\) to get \[y x^2 = 1\]Now, solve for \(x^2\) by dividing both sides by \(y\):\[x^2 = \frac{1}{y}\]

Step 4: Find x in terms of y

Take the square root of both sides to solve for \(x\):\[x = \sqrt{\frac{1}{y}}\]Since we know \(x > 0\), we have \[x = \frac{1}{\sqrt{y}}\]

Step 5: Write the inverse function

Switch \(x\) and \(y\) to express \(y\) as a function of \(x\):\[f^{-1}(x) = \frac{1}{\sqrt{x}}\]This is the inverse function of the given function \(f(x)\) for \(x > 0\).

Step 6: Verify the inverse function

To verify, check that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\):1. \(f(f^{-1}(x)) = f\left(\frac{1}{\sqrt{x}}\right) = \frac{1}{\left(\frac{1}{\sqrt{x}}\right)^2} = x\).2. \(f^{-1}(f(x)) = f^{-1}\left(\frac{1}{x^2}\right) = \frac{1}{\sqrt{\frac{1}{x^2}}} = x\).Since both conditions are satisfied, the inverse is confirmed.

Key Concepts

Function Composition

Function composition is an important concept in mathematics. It involves creating a new function by applying one function to the results of another. If you have two functions, say \( f(x) \) and \( g(x) \), the composition of these two functions is written as \( (f \circ g)(x) \) or \( f(g(x)) \). This means you first apply \( g(x) \) and then apply \( f \) to the result of \( g(x) \).

As a real-world analogy, consider function \( g(x) \) as baking a cake and function \( f(x) \) as frosting it. The composition \( f(g(x)) \) results in a frosted cake.

• To compose two functions, always remember the order: first the inside function, then the outside.
• Function composition is not commutative, meaning \( f(g(x)) \) is generally not the same as \( g(f(x)) \).
• Used for analyzing and transforming inputs within multi-step mathematical operations.

Function composition is key when verifying inverse functions. You must check that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) to ensure that the computed inverse is correct.

Square Root

The square root function represents one of the most fundamental operations in mathematics. If you have a number \( x \), its square root is a number \( y \) such that \( y^2 = x \). Often represented as \( \sqrt{x} \), the square root is essential in solving quadratic equations and finding inverses like in the exercise.

Here are some key points to remember about square roots:

• The square root of a number squared returns the original number (for non-negative numbers), i.e., \( \sqrt{x^2} = x \) provided \( x \geq 0 \).
• In problems involving inverses, taking the square root can be part of the steps to "undo" squaring.
• The square root function is only defined for non-negative numbers in the real number system.

Be cautious about the domain of square roots when dealing with functions. Only positive outputs can safely have their square roots calculated in the inverse process.

Domain of a Function

The domain of a function is the complete set of all possible input values (\( x \)) for which the function is defined. Understanding the domain is crucial as it dictates where the function can operate without issues. In the exercise, the function was \( f(x) = \frac{1}{x^2} \) for \( x > 0 \).

This means

• The function only accepts positive \( x \) values, as \( x^2 \) in the denominator implies division by zero if \( x = 0 \).
• The function \( f(x) \) is undefined for \( x \leq 0 \), establishing \( x > 0 \) as its domain.
• This restriction directly impacts the inverse function's domain, often creating transformations or inverses only applicable within the constraints of the original function's domain.

When calculating inverses, always ensure that the domain and range align with the function's requirements and the real-world context of the problem. This understanding prevents errors when applying functions or presenting results.