Problem 20 - Thomas Calculus

Question

Each of Exercises \(19-24\) gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1} .\) Find a formula for \(f^{-1}\) in each case. $$f(x)=x^{2}, \quad x \leq 0$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Given Function

The function provided is \( f(x) = x^2 \) with the domain restriction \( x \leq 0 \). This means we are only considering the left half of the parabola, where the input (\(x\)) can only be negative or zero.

Step 2: Swap Variables for the Inverse Function

To find the inverse function \( f^{-1}(x) \), we start by swapping \( x \) and \( y \) in the equation \( y = f(x) \). This gives us the equation \( x = y^2 \).

Step 3: Solve for the Original Variable

Now, solve the equation \( x = y^2 \) for \( y \). Since we are dealing with the inverse of a function where \( x \leq 0 \), we take the negative square root to ensure \( y \leq 0 \). Thus, \( y = -\sqrt{x} \).

Step 4: Express the Inverse Function

The final expression for the inverse function is \( f^{-1}(x) = -\sqrt{x} \), valid for \( x \geq 0 \), matching the range of the original function \( f \).

Key Concepts

Inverse Equation Solving

When solving for the inverse of a function, you're essentially reversing the role of inputs and outputs. Think of it like unwinding a puzzle, where the output of the original function becomes the input for the inverse.
In practical steps, this involves the following:

• Start by swapping the dependent and independent variables. For example, if you have a function expressed as \( y = f(x) \), write it as \( x = f(y) \).
• Solve this new equation for \( y \). The result will give you the inverse function \( f^{-1}(x) \).
• Ensure that you are considering the correct domain to reflect back on the range of the original function.

These steps allow you to rewrite the function so that the output-result relationship is reversed, providing an inverse operation.

Domain and Range Restrictions

Both the domain and range are critical when finding and analyzing inverse functions. A domain is the set of permissible inputs, while the range is the set of possible outputs.
When working with inverse functions:

• It's important to remember that the domain of the original function becomes the range of the inverse function.
• Similarly, the range of the original function is the domain of the inverse function.
• In the given exercise, we have a domain restriction where \(x \leq 0\). This is crucial because it tells us that we are only considering part of the function's graph.
• These restrictions ensure that both functions are one-to-one, meaning each input in the domain maps to exactly one output in the range.

Without these considerations, an inverse may not exist, or it might not be a true inverse.

Negative Square Root Function

The negative square root function is essential when dealing with inverse functions of specific parabolic sections, like the one in our exercise. This is because some functions have unique characteristics only in certain sections.
Here's what you need to know:

• When \(f(x) = x^2\) is restricted to \(x \leq 0\), only the left side of the parabola is considered, which impacts the inverse.
• The inverse function is \(-\sqrt{x}\), capturing the need for the inverse's outputs to remain non-positive, mirroring the domain constraints of \(f(x)\).
• By taking the negative square root, you maintain consistency with the original function's range.
• This aspect of the square root function also shows why the inverse is only valid for \(x \geq 0,\) matching the original parabolic range.

Using the negative over the positive square root keeps the function respective to its defined interval.