Problem 45 - Thomas Calculus

Question

If \(f(x)\) is one-to-one, can anything be said about \(g(x)=-f(x) ?\) Is it also one-to-one? Give reasons for your answer.

Step-by-Step Solution

Answer

Yes, \( g(x) = -f(x) \) is also one-to-one because the transformation does not affect the distinctiveness of outputs.

Step 1: Understanding One-to-One Functions

A function \( f(x) \) is considered one-to-one if it maps distinct inputs to distinct outputs. In mathematical terms, for every pair \( x_1 eq x_2 \), it follows that \( f(x_1) eq f(x_2) \). This property ensures that each output is uniquely paired with an input.

Step 2: Define the Function \( g(x) \)

We are given that \( g(x) = -f(x) \). Our task is to determine if \( g(x) \) is also a one-to-one function given that \( f(x) \) is one-to-one.

Step 3: Analyzing the Transformation

The function \( g(x) = -f(x) \) transforms the outputs of \( f(x) \) by taking their negative. We need to check if this transformation affects the one-to-one nature of \( f(x) \).

Step 4: Evaluate One-to-One Property of \( g(x) \)

Suppose \( g(x_1) = g(x_2) \). Then, \(-f(x_1) = -f(x_2)\), which implies that \( f(x_1) = f(x_2) \). Since \( f(x) \) is one-to-one, this means \( x_1 = x_2 \). Therefore, \( g(x) \) is one-to-one.

Key Concepts

Function Transformation

Function transformations involve modifying a function's formula, which results in a change in the output or appearance of the graph of the function. Transformations can include translations, reflections, stretches, and compressions. In the given exercise, the transformation considered is a reflection.

• A reflection transformation, like in our case where f(x) is transformed to g(x) = -f(x), flips the outputs over the horizontal axis. This means each point on the graph of f(x) is mirrored with respect to the x-axis.
• This type of transformation affects the visual representation but does not change the inherent properties of one-to-one functions.

The transformation can be checked by replacing y with -y in the function's formula, hence mapping each output of f(x) to its opposite value.
For a function to preserve its one-to-one nature under transformation, like in the case of reflection, it must ensure that distinct inputs still yield distinct outputs, despite the change in the sign of outputs.

Inverse Functions

Inverse functions are closely related to one-to-one functions. To understand this relationship, it's important to grasp what an inverse function is. A function f(x) has an inverse, denoted as f^-1(x), if it is possible to reverse the function's effect.

• For f(x) to have an inverse, it must be one-to-one. This ensures that every output y comes from a unique input x.
• The inverse function essentially "undoes" the work of f(x), swapping the roles of inputs and outputs.
• Graphically, the inverse is a reflection over the line y = x.

When considering the transformed function g(x) = -f(x), we analyze if it behaves similarly. Transformations like multiplying by a negative number do not alter the one-to-one nature. Hence, if f(x) has an inverse, likely, g(x) will too, but would need to be recomputed as -f^-1(x) if there is a reflection involved.
Understanding how transformations affect inverses is key to appreciating the seamless interchangeability of inputs and outputs in one-to-one functions.

Functional Properties

Functional properties define a function's characteristics and behavior. Understanding these properties is essential for analyzing how a function behaves under various circumstances, including transformations.

• One-to-One Property: This is a central property for determining if a function has an inverse and if transformations like negations preserve uniqueness in mapping.
• Domain and Range: Changing a function's formula can impact its domain and range, although reflections primarily alter the sign or orientation without changing these sets.
• Continuity: Typically, transformations preserve continuity if the original function was continuous.

In our case, ensuring that the one-to-one property holds even after transforming f(x) to g(x) helps ascertain that outputs reflect input changes correctly.
When analyzing functional properties, especially for one-to-one functions, it is crucial to understand how each characteristic is preserved or altered across transformations and potential inverses. This ensures smooth relationships between inputs and outputs are maintained, offering a robust mathematical understanding.