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Chapter 4: Problem 1

Given the function \(f=\\{(1,2),(2,3),(3,4)\\}\) write the set of ordered pairs representing \(f^{-1}\).

Short Answer

Expert verified
The set of ordered pairs for \(f^{-1}\) is \(\{(2,1),(3,2),(4,3)\}\).

Step by step solution

01

- Understand Function Notation

The function notation \(f=\{(1,2),(2,3),(3,4)\}\) means that 1 is mapped to 2, 2 is mapped to 3, and 3 is mapped to 4 by function f.
02

- Define Inverse Function

The inverse function \(f^{-1}\) will reverse these mappings. So the output of \(f\) becomes the input of \(f^{-1}\) and vice versa.
03

- Create Ordered Pairs for Inverse Function

Rewrite each pair in \(f\) by swapping the elements to form \(f^{-1}\). This results in the pairs (2,1), (3,2), and (4,3).
04

- Write the Set of Ordered Pairs

Combine these pairs to represent the inverse function: \(f^{-1}=\{(2,1),(3,2),(4,3)\}\).

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Key Concepts

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

Function Notation
Function notation is a way to describe a function using a specific format.
It typically uses letters like f, g, or h, followed by parentheses, when describing the relationship between input and output values.
In this exercise, we have the function f defined as f={\( \text{(1,2)}, \text{(2,3)}, \text{(3,4)} \)}, which means f maps 1 to 2, 2 to 3, and 3 to 4.
Each pair, such as (1, 2), represents the function f taking an input of 1 and giving an output of 2.
Ordered Pairs
An ordered pair consists of two elements where the order in which they appear is important.
You can think of them as coordinates on a graph.
For example, (1,2) is an ordered pair where 1 is the input and 2 is the output.
The format of ordered pairs in function notation is always (input, output).
It is crucial to remember this because when we talk about the inverse function, we need to swap these pairs.
For the function f={\( \text{(1,2)}, \text{(2,3)}, \text{(3,4)} \)}, the ordered pairs mean converting the function output back to input in the case of the inverse, resulting in (2,1), (3,2), and (4,3).
Mapping
Mapping is the process of associating each element from one set, called the domain, with an element in another set, called the range.
In our exercise, the function f maps inputs from the domain {1, 2, 3} to outputs in the range {2, 3, 4}.
For the inverse function, \(f^{-1}\), the roles of the domain and the range are reversed.
So, \(f^{-1}\) maps the outputs of the original function back to their corresponding inputs, resulting in the pairs: \( \text{(2,1)}, \text{(3,2)}, \text{(4,3)} \).
Understanding mapping helps to visualize how inputs and outputs are connected.

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Most popular questions from this chapter

Fluorodeoxyglucose is a derivative of glucose that contains the radionuclide fluorine- \(18\left({ }^{18} \mathrm{~F}\right) .\) A patient is given a sample of this material containing \(300 \mathrm{MBq}\) of \({ }^{18} \mathrm{~F}\) (a megabecquerel is a unit of radioactivity). The patient then undergoes a PET scan (positron emission tomography) to detect areas of metabolic activity indicative of cancer. After \(174 \mathrm{~min}\), one-third of the original dose remains in the body. a. Write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the radioactivity level \(Q(t)\) of fluorine- 18 at a time \(t\) minutes after the initial dose. b. What is the half-life of \({ }^{18} \mathrm{~F}\) ?

Painful bone metastases are common in advanced prostate cancer. Physicians often order treatment with strontium- \(89\left({ }^{89} \mathrm{Sr}\right)\), a radionuclide with a strong affinity for bone tissue. A patient is given a sample containing \(4 \mathrm{mCi}\) of \({ }^{89} \mathrm{Sr}\). a. If \(20 \%\) of the \({ }^{89} \mathrm{Sr}\) remains in the body after 90 days, write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the amount \(Q(t)\) of radioactivity in the body \(t\) days after the initial dose. b. What is the biological half-life of \({ }^{89} \mathrm{Sr}\) under this treatment?

The half-life of radium-226 is 1620 yr. Given a sample of \(1 \mathrm{~g}\) of radium \(-226,\) the quantity left \(Q(t)\) (in g) after \(t\) years is given by \(Q(t)=\left(\frac{1}{2}\right)^{t / 1620}\) a. Convert this to an exponential function using base \(e\). b. Verify that the original function and the result from (a) yield the same result for \(Q(0), Q(1620),\) and part \(Q(3240) .\) (Note: There may be round-off error.)

a. The populations of two countries are given for January 1,2000 , and for January 1,2010 . Write a function of the form \(P(t)=P_{0} e^{k t}\) to model each population \(P(t)\) (in millions) \(t\) years after January 1, 2000.$$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Population } \\ \text { in 2000 } \\ \text { (millions) } \end{array} & \begin{array}{c} \text { Population } \\ \text { in 2010 } \\ \text { (millions) } \end{array} & \boldsymbol{P}(t)=\boldsymbol{P}_{0} e^{k t} \\ \hline \text { Switzerland } & 7.3 & 7.8 & \\ \hline \text { Israel } & 6.7 & 7.7 & \\ \hline \end{array}$$ b. Use the models from part (a) to predict the population on January \(1,2020,\) for each country. Round to the nearest hundred thousand. c. Israel had fewer people than Switzerland in the year 2000 , yet from the result of part (b), Israel will have more people in the year \(2020 ?\) Why? d. Use the models from part (a) to predict the year during which each population will reach 10 million if this trend continues.

Solve for the indicated variable. \(A=P e^{r t}\) for \(r\) (used in finance)
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