Problem 30 - Precalculus

Question

If \(f\) is odd and invertible, prove that \(f^{-1}\) is also odd.

Step-by-Step Solution

Answer

If \( f \) is odd and invertible, then \( f^{-1} \) is also odd because \( f^{-1}(-y) = -f^{-1}(y) \).

Step 1: Define an Odd Function

A function \( f \) is considered odd if for all \( x \) in its domain, \( f(-x) = -f(x) \). This is the key property of odd functions.

Step 2: Consider the Inverse of \( f \)

Since \( f \) is invertible, every element in the range of \( f \) has a unique pre-image in the domain. The inverse function \( f^{-1} \) satisfies \( f(f^{-1}(y)) = y \) for any \( y \) in the domain of \( f^{-1} \).

Step 3: Express \( f^{-1} \) in Terms of \( f \)

From the property of the inverse function, \( f(f^{-1}(y)) = y \). Now, express \( f^{-1}(-y) \) using \( f \): since \( f \) is odd, \( f(-a) = -f(a) \), so \( f(f^{-1}(-y)) = -y \).

Step 4: Prove \( f^{-1}(-y) = -f^{-1}(y) \)

We need to show that for \( y \) in the domain of \( f^{-1} \), \( f^{-1}(-y) = -f^{-1}(y) \). Since \( f(f^{-1}(-y)) = -y \) and by substituting \( x = f^{-1}(y) \), this implies \( f(f^{-1}(-y)) = f(-x) = -f(x) = -y \). Thus, \( f^{-1}(-y) = -x = -f^{-1}(y) \).

Step 5: Conclude that \( f^{-1} \) is Odd

From Step 4, we have shown that \( f^{-1}(-y) = -f^{-1}(y) \) for all \( y \) in the domain of \( f^{-1} \), satisfying the definition of an odd function. Therefore, \( f^{-1} \) is indeed odd.

Key Concepts

Invertible Function

An invertible function is a special type of function that has a unique quality: every output value maps to one and only one input value. This means that if a function is invertible, we can "reverse" it to find a corresponding input for each output. In more technical terms, a function \( f \) is invertible if there exists another function \( f^{-1} \) such that both \( f(f^{-1}(y)) = y \) and \( f^{-1}(f(x)) = x \) hold true for all applicable \( x \) and \( y \). This property ensures that \( f^{-1} \) effectively "undoes" the action of \( f \), leading to a perfect mapping between inputs and outputs on both sides.

• An invertible function must be bijective, meaning it is both one-to-one and onto.
• One-to-one (injective) means that no two different inputs produce the same output.
• Onto (surjective) means that every possible output is mapped to by some input.

Understanding this concept is crucial when trying to determine the properties of inverse functions.

Inverse Function

The inverse function, denoted as \( f^{-1}(x) \), is essentially a way to "reverse" a given function \( f \). If you have an output from \( f \), the inverse function \( f^{-1} \) gives you back the original input. To find the inverse of a function, you swap the roles of the input and output. Now, for a function to have an inverse, the original function needs to be invertible, as discussed earlier.

• The relationship is described by: \( f(f^{-1}(y)) = y \) and \( f^{-1}(f(x)) = x \).
• If we take any specific value of the original function's range, the inverse function maps it back to the original domain.

Overall, the process of finding an inverse function involves switching inputs and outputs; visually this can be thought of as reflecting the graph of the original function across the line \( y = x \).

Properties of Odd Functions

Odd functions exhibit a unique symmetry. Mathematically, a function \( f \) is called odd if for all \( x \) in its domain, the equation \( f(-x) = -f(x) \) holds true. This symmetry can be visualized as a shape that is perfectly mirrored opposite the origin on a graph.

• This property implies that if you rotate the graph 180 degrees around the origin, it looks the same.
• A practical example would be \( f(x) = x^3 \) or \( f(x) = \sin x \).

This symmetry is key when proving whether a composition, such as an inverse of an odd function, also exhibits this characteristic. Placing emphasis on this symmetry helps in understanding why certain complex functions retain or lose properties like being odd.

Mathematical Proof

Mathematical proof is a logical argument that confirms the truth of a statement or theorem. These proofs help mathematicians verify properties and relationships within functions. In the context of proving whether the inverse of an odd function is also odd, we follow a structured approach involving known properties of functions.

• First, express and exploit known properties. Here, we use the definition of an odd function and properties of the inverse.
• The goal is to manipulate these properties to show that for \( f^{-1}(-y) = -f^{-1}(y) \) holds for all \( y \) in the domain of \( f^{-1} \).

This exercise emphasizes building step-by-step logic, showing the importance of each property used in the proof. Ultimately, by demonstrating that \( f^{-1} \) exhibits symmetrical properties akin to odd functions, you establish it as odd, thereby completing the proof.