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Chapter 4: Problem 59

Graph \(y=2^{x}\) and \(x=2^{y}\) in the same rectangular coordinate system.

Short Answer

Expert verified
Plot exponential function \(y=2^{x}\) by picking suitable \(x\) values. Then calculate corresponding y values using the relation \(y=2^{x}\). For the inverse function, rewrite the equation as \(y=\log_2{x}\) and pick suitable values for \(x\), then calculate \(y\) values using the relation \(y=\log_2{x}\). The resulting plot should illustrate that the two graphs are reflections of each other over the line \(y=x\).

Step by step solution

01

Understand and Plot the Exponential Function \(y=2^{x}\)

The function \(y=2^{x}\) is an exponential function. Begin by choosing values for \(x\) and calculating corresponding \(y\). Choose values that are easy to calculate such as -2, -1, 0, 1, and 2. Plot these points to create a smooth curve for the function \(y=2^{x}\).
02

Understand the Inverse Function \(x=2^{y}\)

The equation \(x=2^{y}\) can be rewritten as \(y=\log_2{x}\), which is the inverse function of \(y=2^{x}\). Here, \(y\) can only take values when \(x\) is positive. So, select values of \(x\) ranging from 0.25, 1, 2, 4 onward, and for each \(x\), calculate the corresponding value of \(y\) using the expression \(y=\log_2{x}\).
03

Plot the Inverse Function \(x=2^{y}\)

Plot the points calculated in Step 2 on the same rectangular coordinate system. You will find that \(x\) is always positive in this case.

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Most popular questions from this chapter

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ \log (x-15)+\log x=2 $$

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ 3^{x}=2 x+3 $$

Explaining the Concepts One problem with all exponential growth models is that nothing can grow exponentially forever. Describe factors that might limit the size of a population.

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. By \(2009,\) the models indicate that Canada's population will exceed Uganda's by approximately 2.8 million.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After 100 years, a population whose growth rate is \(3 \%\) will have three times as many people as a population whose growth rate is \(1 \%\)
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