Problem 49 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=x^{4}, \quad x \geq 0\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = x^{1/4}\), with \(x \geq 0\).

Step 1: Verify the Function is One-to-One

To find the inverse, first ensure the function is one-to-one. For \(f(x) = x^4\) and \(x \geq 0\), the function is strictly increasing, so it is one-to-one in this domain.

Step 2: Replace f(x) with y

Let \(y = f(x)\). Thus, we have the equation \(y = x^4\). This helps set up the equation for finding the inverse.

Step 3: Swap x and y

Swap \(x\) and \(y\) to begin solving for the inverse function. Thus, \(x = y^4\).

Step 4: Solve for y in Terms of x

To solve for \(y\), take the fourth root of both sides because \(x = y^4\). This gives us \(y = \sqrt[4]{x}\) or \(y = x^{1/4}\).

Step 5: Write the Inverse Function

The inverse function is \(f^{-1}(x) = x^{1/4}\) where \(x \geq 0\). Remember to specify the domain of this inverse function.

Key Concepts

One-to-One Functions

A function is one-to-one, or injective, if it assigns distinct output values to distinct input values. This means that no two different inputs produce the same result. Knowing whether a function is one-to-one is crucial when finding an inverse because only one-to-one functions have inverses that are also functions.

• To determine if a function is one-to-one, use the Horizontal Line Test: If any horizontal line crosses the graph of the function at most once, the function is one-to-one.
• In our example, the function \(f(x) = x^4\) with \(x \geq 0\) is strictly increasing. This implies it passes the Horizontal Line Test, confirming it is one-to-one on this domain.

Recognizing a function as one-to-one ensures that its inverse will uniquely map inputs back to outputs, maintaining the fundamental property of a function.

Function Domains

The domain of a function consists of all possible input values (x-values) for which the function is defined. In simpler terms, it is the complete set of inputs we can use without breaking any mathematical rules.

• Domains can be restricted for various reasons, such as avoiding division by zero or ensuring a real number result for square roots.
• For the function \(f(x) = x^4\), the domain is restricted to \(x \geq 0\). This restriction ensures the function is one-to-one within this region, which is necessary for finding its inverse.

Understanding the domain is essential when calculating inverses as it helps define the range of the inverse function. For example, since the original function's domain is \(x \geq 0\), the inverse function \(f^{-1}(x) = x^{1/4}\) will also have the domain \(x \geq 0\), ensuring that it works seamlessly.

Function Composition

Function composition involves applying one function to the results of another. This operation can demonstrate the relationship between a function and its inverse.

• If you compose a function and its inverse, you should obtain the identity function, meaning \( (f \circ f^{-1})(x) = x \) and \( (f^{-1} \circ f)(x) = x \). This property confirms that you have correctly identified an inverse function.
• To illustrate: if \(f(x) = x^4\) for \(x \geq 0\) and its inverse \(f^{-1}(x) = x^{1/4}\) for \(x \geq 0\), then \(f(f^{-1}(x)) = (x^{1/4})^4 = x\) and \(f^{-1}(f(x)) = (x^4)^{1/4} = x\).

Function composition is key to verifying the accuracy of an inverse function. By understanding and applying composition, you ensure the inverse relationship holds true throughout the function's domain.