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Chapter 5: Problem 22

Graph the pair of functions on the same set of axes. $$y=(1 / 3)^{x} ; y=3^{x}$$

Short Answer

Expert verified
Graph the functions to see \( y = 3^x \) increase and \( y = (1/3)^x \) decrease, both passing through \( (0, 1) \).

Step by step solution

01

Understand the Functions

The two functions given are exponential functions. The first function is \( y = \left( \frac{1}{3} \right)^x \), and the second function is \( y = 3^x \). Both functions have a base of 3; however, the first function has a base that is the reciprocal of the base in the second function.
02

Identify Key Properties of Each Function

For \( y = 3^x \), as \( x \) increases, \( y \) increases because 3 is greater than 1. This function is an increasing exponential function. For \( y = (1/3)^x \), as \( x \) increases, \( y \) decreases because 1/3 is less than 1. This function is a decreasing exponential function.
03

Find Points to Plot

Calculate values for key points. For \( y = 3^x \), use points like \( x = -2, -1, 0, 1, 2 \) to calculate \( y \) as \( 1/9, 1/3, 1, 3, 9 \). Similarly, for \( y = (1/3)^x \), use the same \( x \)-values to find \( y \) as \( 9, 3, 1, 1/3, 1/9 \).
04

Plot the Functions on Axes

Begin by drawing the horizontal axis \( (x) \) and the vertical axis \( (y) \). Plot the points for each function: For \( y = 3^x \), plot \((-2, 1/9), (-1, 1/3), (0, 1), (1, 3), (2, 9)\). For \( y = (1/3)^x \), plot \((-2, 9), (-1, 3), (0, 1), (1, 1/3), (2, 1/9)\).
05

Draw the Curves

Using the plotted points, draw a smooth curve through the points for each function. The curve for \( y = 3^x \) should rise steeply as \( x \) increases, while the curve for \( y = (1/3)^x \) should fall steeply as \( x \) increases. Both curves will intersect at \( (0, 1) \).
06

Compare and Analyze the Graphs

Observe that \( y = 3^x \) is an increasing function and \( y = (1/3)^x \) is a decreasing function. Both are reflections of each other about the line \( y = x \) due to their reciprocal nature.

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Key Concepts

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

Graphing Functions
Exponential functions like \( y = \left( \frac{1}{3} \right)^x \) and \( y = 3^x \) are fascinating to graph because they each exhibit unique growth behavior. These functions help us visualize how changing the base influences the function's direction of growth. A typical graphing process starts with drawing the axes:
  • The horizontal axis represents the \( x \)-values, while the vertical axis represents the corresponding \( y \)-values.
  • For \( y = 3^x \), as \( x \) increases, the \( y \)-value grows rapidly indicating an exponentially increasing graph.
  • Conversely, \( y = \left( \frac{1}{3} \right)^x \) decreases as \( x \) increases since the base \( \frac{1}{3} \) is less than 1.
Plotting these graphs involves marking key points on the axes for each function and then confidently drawing a smooth curve through them. Remember, the curve for \( y = 3^x \) points upward while for \( \left( \frac{1}{3} \right)^x \) it slopes downward.
Inverse Functions
In the world of functions, an inverse reflects one function across the line \( y = x \), swapping each function's \( x \)- and \( y \)-values. In our exercise, \( y = 3^x \) and \( y = \left( \frac{1}{3} \right)^x \) are related in a special way. They serve as inverses of each other by flipping the base in the expression:
  • The base of the increasing function \( y = 3^x \) is inverted to create the decreasing function \( \left( \frac{1}{3} \right) \).
  • Because each function is the reciprocal of the other, their graphs symmetrically reflect over the line \( y = x \).
Understanding this concept is essential because it reveals the symmetry present in exponential behavior, illustrating how expanding in one scenario results in shrinking in another.
Plotting Points
Plotting points is a practical method to construct graphs of functions accurately. In our scenario, you need key values to plot exponential functions \( y = 3^x \) and \( y = \left( \frac{1}{3} \right)^x \):
  • Select specific \( x \)-values like \(-2, -1, 0, 1, 2\) to compute \( y \).
  • For \( y = 3^x\), the computed \( y \)-values are \( \frac{1}{9}, \frac{1}{3}, 1, 3, 9 \).
  • For \( y = \left( \frac{1}{3} \right)^x\), the points flip, giving \( 9, 3, 1, \frac{1}{3}, \frac{1}{9} \).
Plotting these points on the graph provides a skeleton that guides placing your curves accurately. Proper plot points ensure you capture the slope and direction of growth or decay, building a reliable visual representation of these mathematical functions.

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Most popular questions from this chapter

Exercises \(55-60\) introduce a model for population growth that takes into account limitations on food and the environment. This is the logistic growth model, named and studied by the nineteenth century Belgian mathematician and sociologist Pierre Verhulst. (The word "logistic" has Latin and Greek origins meaning "calculation" and "skilled in calculation," respectively. However, that is not why Verhulst named the curve as he did. See Exercise 56 for more about this.) In the logistic model that we "I study, the initial population growth resembles exponential growth. But then, at some point owing perhaps to food or space limitations, the growth slows down and eventually levels off, and the population approaches an equilibrium level. The basic equation that we'll use for logis- tic growth is where \(\mathcal{N}\) is the population at time \(t, P\) is the equilibrium population (or the upper limit for population), and a and b are positive constants. $$\mathcal{N}=\frac{P}{1+a e^{-b t}}$$ The following figure shows the graph of the logistic function \(\mathcal{N}(t)=4 /\left(1+8 e^{-t}\right) .\) Note that in this equation the equilibrium population \(P\) is 4 and that this corresponds to the asymptote \(\mathcal{N}=4\) in the graph. (a) Use the graph and your calculator to complete the following table. For the values that you read from the graph, estimate to the nearest \(0.25 .\) For the calculator values, round to three decimal places. (b) As indicated in the graph, the line \(\mathcal{N}=4\) appears to be an asymptote for the curve. Confirm this empirically by computing \(\mathcal{N}(10), \mathcal{N}(15),\) and \(\mathcal{N}(20) .\) Round each answer to eight decimal places. (c) Use the graph to estimate, to the nearest integer, the value of \(t\) for which \(\mathcal{N}(t)=3\) (d) Find the exact value of \(t\) for which \(\mathcal{N}(t)=3 .\) Evaluate the answer using a calculator, and check that it is consistent with the result in part (c). TABLE AND GRAPH CANT COPY

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Solve the inequalities. Where appropriate, give an exact answer as well as a decimal approximation. $$3\left(2-0.6^{x}\right) \leq 1$$

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