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Chapter 0: Problem 25

In Exercises 25 and 26, does the function have an inverse function? $$ \begin{array}{|l|r|r|r|r|r|r|} \hline x & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & -2 & 1 & 2 & 1 & -2 & -6 \\ \hline \end{array} $$

Short Answer

Expert verified
No, the function does not have an inverse function.

Step by step solution

01

Identify the y-values in the function

First, look at the function and especially identify y-values, in this case, they are -2, 1, 2, 1, -2, -6.
02

Examine the y-values for repetition

Examine identified y-values to ensure they are unique i.e., each y-value is linked to a different x-value. Notice that the y-values -2 and 1 are repeated which means that there are different x-values which produce the same y-values.
03

Conclusion

Consequently, the function is not a one-to-one function. Hence, this function does not have an inverse function according to the definition of inverse function.

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Key Concepts

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

Understanding One-to-One Functions
A one-to-one function is a special kind of function where each y-value is paired with exactly one unique x-value. In mathematical terms, if a function \( f(x) \) is one-to-one, it means for any two values \( x_1 \) and \( x_2 \), \( f(x_1) = f(x_2) \) implies that \( x_1 = x_2 \). This ensures that every output is distinct and corresponds to just one input.

If a function repeats any y-values for different x-values, it ceases to be one-to-one. Only one-to-one functions can have inverses because each input maps to a unique output. In the example function provided, since some y-values repeat, this function is not one-to-one.
Exploring Function Analysis
Function analysis involves examining how a function behaves. It includes studying the relationship between x and y values and observing the pattern or rule connecting them. A critical aspect of function analysis is determining whether a function is one-to-one by analyzing the function's output values.

By closely examining the y-values of a function, you can determine the potential for an inverse function. If a function yields the same y-value from different x-values, that indicates the function is not one-to-one and will not have an inverse function. Function analysis provides deep insight by ensuring the distinctiveness required for inverse computations.
Repetition of Y-Values and Its Impact
When y-values repeat in a function, it signifies that different x-values are producing the same y-value. This repetition has an important implication: it indicates that the function is not one-to-one.

Let's consider another example. If a function maps \( x = 1 \) to \( y = 3 \) and \( x = 2 \) also to \( y = 3 \), then y-value 3 appears twice. To determine whether y-values repeat:
  • Look for duplicate numbers in the y-value list.
  • Check if these repeats come from different x-values.
Repetitive y-values prevent a function from having an inverse because it lacks the one-to-one mapping necessary for such an existence.
Defining Inverse Functions
An inverse function is essentially a reverse of the original function. If a function \( f(x) \) maps x-values to y-values, its inverse function \( f^{-1}(y) \) maps the y-values back to their original x-values.

For a function to have an inverse, it must be one-to-one. This ensures that each y-value has precisely one corresponding x-value, creating a perfect pair for reversal. Without this unique pair, the inverse could not uniquely map each y back to its x, leading to ambiguity.
  • This requirement ensures the uniqueness of mapping in both directions: x to y and y back to x.
  • It provides the ideal condition for creating inverses, allowing us to reverse the function flawlessly.
Only functions without repeated y-values (or one-to-one functions) can have legitimate inverses.

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Most popular questions from this chapter

Path of a Ball The height \(y\) (in feet) of a baseball thrown by a child is $$ y=-\frac{1}{10} x^{2}+3 x+6 $$ where \(x\) is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.)

Digital Camera Sales The factory sales \(f\) (in millions of dollars) of digital cameras in the United States from 1998 through 2003 are shown in the table. The time (in years) is given by \(t\), with \(t=8\) corresponding to 1998 . (Source: Consumer Electronincs Association) $$ \begin{array}{|c|c|} \hline \text { Year, } t & \text { Sales, } f(t) \\ \hline 8 & 519 \\ 9 & 1209 \\ 10 & 1825 \\ 11 & 1972 \\ 12 & 2794 \\ 13 & 3421 \\ \hline \end{array} $$ (a) Does \(f^{-1}\) exist? (b) If \(f^{-1}\) exists, what does it represent in the context of the problem? (c) If \(f^{-1}\) exists, find \(f^{-1}(1825)\). (d) If the table was extended to 2004 and if the factory sales of digital cameras for that year was \(\$ 2794\) million, would \(f^{-1}\) exist? Explain.

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ g(x)=\frac{x}{8} $$

Temperature The table shows the temperature \(y\) (in degrees Fahrenheit) of a certain city over a 24-hour period. Let \(x\) represent the time of day, where \(x=0\) corresponds to \(6 \mathrm{~A}\).M. $$ \begin{array}{|c|c|} \hline \text { Time, } \boldsymbol{x} & \text { Temperature, } \boldsymbol{y} \\\ \hline 0 & 34 \\ 2 & 50 \\ 4 & 60 \\ 6 & 64 \\ 8 & 63 \\ 10 & 59 \\ 12 & 53 \\ 14 & 46 \\ 16 & 40 \\ 18 & 36 \\ 20 & 34 \\ 22 & 37 \\ 24 & 45 \\ \hline \end{array} $$ A model that represents these data is given by \(y=0.026 x^{3}-1.03 x^{2}+10.2 x+34, \quad 0 \leq x \leq 24 .\) (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? (c) Use the graph to approximate the times when the temperature was increasing and decreasing. (d) Use the graph to approximate the maximum and minimum temperatures during this 24 -hour period. (e) Could this model be used to predict the temperature for the city during the next 24 -hour period? Why or why not?

True or False? Determine whether the statement is true or false. Justify your answer. It is possible for an odd function to have the interval \([0, \infty)\) as its domain.
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