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Chapter 4: Problem 14

Graph both functions on one set of axes. $$f(x)=\left(\frac{2}{3}\right)^{x} \quad \text { and } \quad g(x)=\left(\frac{4}{3}\right)^{x}$$

Short Answer

Expert verified
Exponential function \( f(x) \) decreases, while \( g(x) \) increases; both pass through \((0,1)\).

Step by step solution

01

Understanding the Functions

Both functions are exponential functions of the form \( a^x \), where \( a > 0 \). For the function \( f(x) = \left(\frac{2}{3}\right)^x \), the base is less than 1, indicating that it is a decreasing function. For \( g(x) = \left(\frac{4}{3}\right)^x \), the base is greater than 1, suggesting it is an increasing function.
02

Identifying Intercepts

Both functions will pass through the point \((0, 1)\), since any number raised to the power of zero is 1, i.e., \( f(0) = \left(\frac{2}{3}\right)^{0} = 1 \) and \( g(0) = \left(\frac{4}{3}\right)^{0} = 1 \).
03

Plotting f(x) = (2/3)^x

To graph \( f(x) = \left(\frac{2}{3}\right)^x \), calculate a few points, such as \( f(1) = \frac{2}{3} \), \( f(-1) = \frac{3}{2} \), and \( f(2) = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \). Plot these and note the curve is falling as \( x \) increases.
04

Plotting g(x) = (4/3)^x

To graph \( g(x) = \left(\frac{4}{3}\right)^x \), calculate a few points, such as \( g(1) = \frac{4}{3} \), \( g(-1) = \frac{3}{4} \), and \( g(2) = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \). Plot these and note the curve is rising as \( x \) increases.
05

Sketching the Graph

Draw both the decreasing curve of \( f(x) \) and the increasing curve of \( g(x) \) on the same set of axes. Use the plotted points and their behaviors (decreasing for \( f(x) \), increasing for \( g(x) \)) to assist in sketching smooth curves.

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Key Concepts

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

Graphing Functions
Graphing functions is like creating a visual representation of a mathematical equation. It helps us understand how the function behaves over different values of the variable. When dealing with exponential functions such as \( f(x) = \left(\frac{2}{3}\right)^x \) and \( g(x) = \left(\frac{4}{3}\right)^x \), we need to be aware of particular characteristics that define their graphs.

For these functions:
  • The function \( f(x) = \left(\frac{2}{3}\right)^x \) will have a curve that starts high on the y-axis and declines as \( x \) increases.
  • The function \( g(x) = \left(\frac{4}{3}\right)^x \) will start lower and ascend as \( x \) becomes larger.
To graph them, begin by calculating several key points:
  • For \( f(x) \), calculate values such as \( f(0) = 1 \), \( f(1) = \frac{2}{3} \), and \( f(-1) = \frac{3}{2} \).
  • For \( g(x) \), check \( g(0) = 1 \), \( g(1) = \frac{4}{3} \), and \( g(-1) = \frac{3}{4} \).
Plotting these points on the Cartesian plane and drawing smooth curves through them accomplishes the task of graphing these functions.
Increasing and Decreasing Functions
An increasing function means that as the input value, \( x \), increases, the output value, \( y \), also increases. In contrast, a decreasing function has a y-value that declines as x gets larger.

For exponential functions, the behavior depends largely on the base of the equation:
  • When the base is greater than 1, the function is increasing. This is the case for \( g(x) = \left(\frac{4}{3}\right)^x \). Its graph rises upwards to the right.
  • When the base is between 0 and 1, the function is decreasing, like \( f(x) = \left(\frac{2}{3}\right)^x \). This graph descends to the right.
It's essential to recognize these behaviors when analyzing or sketching graphs, as it affects the overall shape and direction of the curve. Understanding whether a function is increasing or decreasing helps us predict how the function behaves over an interval.
Intercepts of Functions
Intercepts are points where the graph of a function crosses the axes. For most functions, one primary intercept is the y-intercept, where the function crosses the y-axis. The x-intercept, if it exists, is where the function crosses the x-axis.

For exponential functions like \( f(x) = \left(\frac{2}{3}\right)^x \) and \( g(x) = \left(\frac{4}{3}\right)^x \):
  • Both have a y-intercept at \( (0, 1) \). That鈥檚 because any non-zero number raised to the 0 power is 1, which applies to both functions.
  • These particular exponential functions will not have x-intercepts because their values never actually become zero. They asymptotically approach the x-axis but never touch or cross it.
Understanding intercepts is crucial as they serve as critical reference points in graphing and analyzing many different types of functions.

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