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Chapter 1: Problem 49

Give an example of a line in the coordinate plane that is not the graph of any function.

Short Answer

Expert verified
The example of a line in the coordinate plane that is not the graph of any function is the vertical line \(x = 2\), as it fails the vertical line test, having multiple y-values for a single x-value.

Step by step solution

01

Find a line that fails the vertical line test

To find a line that fails the vertical line test, we need a line that intersects multiple points for a single x-value. A simple example of such a line is a vertical line, which is parallel to the y-axis and has the same x-value for all its points.
02

Determine the equation of a vertical line

Vertical lines have the general equation x = k, where k is a constant. This equation shows that the x-value remains constant, while the y-value can take any value. Therefore, our required example of the line can be any vertical line x = k.
03

Choose a value for k

Let's choose a value for k in the equation x = k. A simple choice is k = 2. Therefore, our example of a line in the coordinate plane that is not the graph of any function is the vertical line x = 2.

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Most popular questions from this chapter

For each of the functions \(f\) given. (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part (c) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I\) (recall that \(I\) is the function defined by \(I(x)=x\) ). \(f(x)=3 x+5\)

Suppose \(f\) is a one-to-one function. Explain why the inverse of the inverse of \(f\) equals \(f\). In other words, explain why $$\left(f^{-1}\right)^{-1}=f$$

Suppose \(f\) is a function and a function \(g\) is defined by the given expression. (a) Write \(g\) as the composition of \(f\) and one or two linear functions. (b) Describe how the graph of \(g\) is obtained from the graph of \(f\). \( g(x)=-5 f\left(-\frac{4}{3} x\right)-8\)

Suppose you are exchanging cur. rency in the London airport. The currency exchange service there only makes transactions in which one of the two currencies is British pounds, but you want to exchange dollars for Euros. Thus you first need to exchange dollars for British pounds, then exchange British pounds for Euros. At the time you want to make the exchange, the function \(f\) for exchanging dollars for British pounds is given by the formula \(f(d)=0.66 d-1\) and the function \(g\) for exchanging British pounds for Euros is given by the formula \(g(p)=1.23 p-2\) The subtraction of 1 or 2 in the number of British pounds or Euros that you receive is the fee charged by the currency exchange service for each transaction. How many Euros would you receive for exchanging $$\$ 100$$ after going through this two-step exchange process?

Suppose \(f\) and \(g\) are functions, each with domain of four numbers, with \(f\) and \(g\) defined by the tables below: $$\begin{array}{c|c}x & f(x) \\\\\hline 1 & 4 \\\2 & 5 \\\3 & 2 \\\4 & 3\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & 3 \\\3 & 2 \\\4 & 4 \\\5 & 1\end{array}$$ Give the table of values for \(g \circ g^{-1}\).
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