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Chapter 5: Problem 6

Decide whether each function is one-to-one. Do not use a calculator. $$f(x)=-\sqrt{100-x^{2}}$$

Short Answer

Expert verified
The function is not one-to-one since different inputs can result in the same output.

Step by step solution

01

Understand the Definition of a One-to-One Function

A function is one-to-one if each output value is associated with exactly one input value. Alternatively, if two inputs yield the same output, then the function is not one-to-one.
02

Analyze the Function Algebraically

Consider the given function \( f(x) = -\sqrt{100-x^2} \). The function involves a square root, which typically outputs two possible values (positive and negative). However, here it is negated, implying all outputs are non-positive.
03

Determine the Function's Range

The expression \( \sqrt{100-x^2} \) yields values from 0 to 10 for \(-10 \leq x \leq 10\). Given this range, \( f(x) = -\sqrt{100-x^2} \) yields values from -10 to 0.
04

Check for Differing Inputs with Same Outputs

Consider specific values, such as \( x = 6 \) and \( x = -6 \). Both yield \( f(6) = f(-6) = -\sqrt{100-6^2} = -\sqrt{64} = -8 \). Thus, different inputs produce the same output, showing \( f(x) \) is not one-to-one.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Analysis
When analyzing functions, especially for determining whether they are one-to-one, it's essential to understand the basic principles behind the function's behavior. A one-to-one function is characterized by each output being paired with one unique input. This ensures that no two different inputs can produce the same output.

To check if a function is one-to-one, consider simple tests:
  • Graphical observation: A horizontal line should intersect the graph of the function at most once.
  • Algebraic analysis: Using logic by assuming different inputs produce the same output and solving for contradictions.
By understanding these principles, we can effectively assess a function's nature and its uniqueness in terms of input-output relationships.
Algebraic Analysis
To determine if a function like \( f(x)=-\sqrt{100-x^2} \) is one-to-one, we need to perform an algebraic analysis. This involves examining the function's structure and behavior. The given function involves a square root, indicating positive and negative possible values. However, it is negated here, which means all outputs are non-positive values.

Consider the following:
  • The term \( \sqrt{100-x^2} \) represents a circle equation in a different form.
  • Negating the square root ensures all outputs range from 0 downwards, potentially leading to identical outputs from different inputs.
This algebraic approach highlights key attributes that may affect the one-to-one nature of a function.
Domain and Range
The concepts of domain and range are crucial in understanding any function thoroughly. The **domain** refers to all possible values for which a function is defined, while the **range** is the set of potential values the function can output.

For the function \( f(x) = -\sqrt{100-x^2} \):
  • **Domain**: Determined by the condition under the square root, \( 100-x^2 \geq 0 \), simplifying to \(-10 \leq x \leq 10\).
  • **Range**: Calculated from the expression \( \sqrt{100-x^2} \), which outputs numbers from 0 to 10, but given the negative square root, these values are inverted from -10 to 0.
Understanding the domain and range helps visualize the boundaries within which the function operates.
Square Root Functions
Square root functions, such as \( f(x) = -\sqrt{100-x^2} \), possess unique characteristics due to the square root component. They are generally restricted by both their domain and the inherent positive and negative values that roots suggest.

Key aspects include:
  • Even though square roots yield both positive and negative solutions, the extra negative sign in the function affects the nature of the solution.
  • This particular function relates to a semicircle's negative half due to the format \( \sqrt{100-x^2} \).
  • Despite limited ranges (from circular restrictions), square roots denote a symmetry which could be useful during analysis.
These elements define challenges and special considerations when assessing the one-to-one nature of functions including square roots.

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