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Chapter 4: Problem 6

In \(3-6,\) each set represents a function. a. What is the domain of each function? b. What is the range of each function? c.Is the function one-to-one? $$ \\{(0,3),(-1,5),(-2,7),(-3,9),(-4,11)\\} $$

Short Answer

Expert verified
Domain: \(\{0, -1, -2, -3, -4\}\); Range: \(\{3, 5, 7, 9, 11\}\); One-to-One: Yes.

Step by step solution

01

Identify Domain

The domain of a function is the set of all possible input values (usually \(x\)-coordinates) that the function can accept. Looking at the set, the input values are \(0, -1, -2, -3, -4\). Thus, the domain is \(\{0, -1, -2, -3, -4\}\).
02

Identify Range

The range of a function is the set of all possible output values (usually \(y\)-coordinates) that the function can produce. From the set, the output values are \(3, 5, 7, 9, 11\). Hence, the range is \(\{3, 5, 7, 9, 11\}\).
03

Determine if Function is One-to-One

A function is one-to-one if each input value pairs with a unique output value and each output value is paired with a unique input value (no repeated \(y\) values for different \(x\) values). In this set, each input \(x\) value pairs with a unique output \(y\) value, and there are no repeated \(y\) values for different \(x\) values. Therefore, this function is one-to-one.

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Key Concepts

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

Domain
The domain of a function refers to the complete set of possible input values, often represented as the set of all possible values of the independent variable, usually denoted as \( x \). In a function, these input values correspond to the first elements of ordered pairs. For the given set of ordered pairs \( \{(0,3),(-1,5),(-2,7),(-3,9),(-4,11)\} \), the inputs are \( 0, -1, -2, -3, \) and \( -4 \). Thus, the domain is the set \( \{0, -1, -2, -3, -4\} \).
  • It's important to identify the domain when working with functions, as it defines the scope of inputs the function can accept.
  • The domain determines the limitations or breadth of the function's applicability.
Range
The range of a function consists of all possible outputs or results provided by the function, often represented as the potential values of the dependent variable, usually denoted as \( y \). In a function defined by a set of ordered pairs, the range corresponds to the second elements of those pairs. Referring back to our set \( \{(0,3),(-1,5),(-2,7),(-3,9),(-4,11)\} \), the outputs are \( 3, 5, 7, 9, \) and \( 11 \). Therefore, the range is \( \{3, 5, 7, 9, 11\} \).
  • Understanding the range helps to know what outputs we can expect from the function for a given domain.
  • The range allows us to explore the function's behavior in response to its domain.
One-to-One Function
A one-to-one function is a special type of function where each input value is associated with a unique output value, and each output value corresponds to a unique input value. This ensures that there are no repeated output values (no two distinct input values have the same output).
Consider the function represented by the set \( \{(0,3),(-1,5),(-2,7),(-3,9),(-4,11)\} \). In this set, every input value maps to a distinct output value, and there are no duplicate output values for different input values. Therefore, this function is classed as one-to-one.
  • A one-to-one function ensures that the function is invertible, meaning it has an inverse function that can reverse the mapping.
  • This concept is crucial in various fields of mathematics, including algebra and calculus, for solving equations and analyzing functions.

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Most popular questions from this chapter

In \(11-18 :\) a. Find \(h(x)\) when \(h(x)=g(f(x)) .\) b. What is the domain of \(h(x) ?\) c. What is the range of \(\mathrm{h}(x) ?\) d. Graph \(\mathrm{h}(x)\) $$ \mathrm{f}(x)=3 x-1, \mathrm{g}(x)=\frac{1}{3}(x+1) $$

If \(\mathrm{g}=\\{(x, y) : y=7-x\\},\) find \(\mathrm{g}^{-1}\) if it exists. Is it possible for a function to be its own inverse?

Aaron can ride his bicycle to school at an average rate that is three times that of the rate at which he can walk to school. How does the time that it takes Aaron to ride to school compare with the time that it takes him to walk to school?

Give an example of a function g for which \(2 \mathrm{g}(x) \neq \mathrm{g}(2 x) .\) Give an example of a function \(\mathrm{f}\) for which \(2 \mathrm{f}(x)=\mathrm{f}(2 x)\)

In \(11-16,\) determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function. $$ \\{(1,4),(2,7),(1,10),(4,13)\\} $$
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