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Chapter 5: Problem 10

(Graphing program required.) Graph the functions \(f(x)=6(0.7)^{x}\) and \(g(x)=6(1.3)^{x}\) on the same grid. Supply the symbol \(<,>,\) or \(=\) in the blank that would make the statement true. a. \(f(0)\)_____\(g(0)\) b. \(f(5)\)_____\(g(5)\) c. \(f(-5)\)_____\(-g(-5)\) d. As \(x \rightarrow+\infty, f(x)\)_____\(g(x)\) e. As \(x \rightarrow-\infty, f(x)\)_____\(g(x)\)

Short Answer

Expert verified
a. = , b. < , c. > , d. < , e. >

Step by step solution

01

Define the Functions

Define the given functions: Function 1: \[f(x) = 6(0.7)^x\] Function 2: \[g(x) = 6(1.3)^x\]
02

Evaluate Both Functions at x = 0

Calculate the value of each function when \(x = 0\): \[ f(0) = 6(0.7)^0 = 6 \times 1 = 6 \] \[g(0) = 6(1.3)^0 = 6 \times 1 = 6 \] Since \(f(0) = g(0)\), place an \(=\) in the blank.
03

Evaluate Both Functions at x = 5

Calculate the value of each function when \(x = 5\): \[f(5) = 6(0.7)^5 \approx 6 \times 0.16807 \approx 1.00842\] \[g(5) = 6(1.3)^5 \approx 6 \times 3.71293 \approx 22.27758\] Since \(f(5) < g(5)\), place an \(<\) in the blank.
04

Evaluate Both Functions at x = -5

Calculate the value of each function when \(x = -5\): \[f(-5) = 6(0.7)^{-5} \approx 6 \times 6.80744 \approx 40.84464\] \[g(-5) = 6(1.3)^{-5} \approx 6 \times 0.28431 \approx 1.70586\] Now calculate \(-g(-5)\): \[-g(-5) \approx -1.70586\] Since \(f(-5) > -g(-5)\), place an \(>\) in the blank.
05

Determine the Behavior as x Approaches Positive Infinity

Compare the behavior of both functions as \(x \rightarrow +\infty\): \[f(x) = 6(0.7)^x \rightarrow 0\] \[g(x) = 6(1.3)^x \rightarrow \infty\] Since \(f(x) \rightarrow 0\) and \(g(x) \rightarrow \infty\), place an \(<\) in the blank.
06

Determine the Behavior as x Approaches Negative Infinity

Compare the behavior of both functions as \(x \rightarrow -\infty\): \[f(x) = 6(0.7)^x \rightarrow \infty\] \[g(x) = 6(1.3)^x \rightarrow 0\] Since \(f(x) \rightarrow \infty\) and \(g(x) \rightarrow 0\), place an \(>\) in the blank.

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Key Concepts

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

Function Evaluation
Function evaluation is the process of finding the value of a function at a particular point. For this exercise, we evaluated the functions at specific values of x.
When evaluating the functions at x = 0, we found that both functions have the same value: 6. This is because any number raised to the power of 0 is 1. Thus, both functions equal 6 at this point.
The process involves substituting the given x-value into the function's equation and performing the arithmetic. For instance:
  • f(0) = 6(0.7)^0 = 6
  • g(0) = 6(1.3)^0 = 6
This simple substitution helps us understand how the functions behave at different points.
Graphing
Graphing functions provides a visual representation of their behavior over a range of x-values. In this problem, we graph the functions f(x) = 6(0.7)^x and g(x) = 6(1.3)^x on the same grid.
This visual aid helps us understand the differences between the two functions over different intervals of x. For example:
  • f(x) = 6(0.7)^x starts at 6 when x = 0 and decreases as x increases.
  • g(x) = 6(1.3)^x also starts at 6 when x = 0 but increases as x increases.
By graphing these functions, we can see that f(x) decreases rapidly while g(x) increases rapidly. This visualization assists in understanding their behaviors at positive and negative infinities.
Limits
Limits help us understand the behavior of functions as x approaches infinity or negative infinity. For exponential functions like those in our exercise, the limit as x approaches infinity and negative infinity shows stark differences.
As x goes to positive infinity:
  • f(x) = 6(0.7)^x approaches 0 since 0.7 is a fraction, and raising it to larger and larger powers makes it smaller.
  • g(x) = 6(1.3)^x approaches infinity because 1.3 is greater than 1, and it grows larger as it's raised to higher powers.
As x approaches negative infinity:
  • f(x) approaches infinity because (0.7)^{-x} grows larger as x becomes more negative.
  • g(x) approaches 0 because (1.3)^{-x} makes the base smaller as x becomes more negative.
Understanding limits is crucial in determining the long-term behavior of functions.
Inequalities
In this exercise, we compare the values of the functions using inequalities. Inequalities help us determine which function's value is greater at specific points.
For example:
  • At x = 0, both functions are equal (f(0) = g(0)).
  • At x = 5, f(5) is much smaller than g(5) (since 0.7^5 is much smaller than 1.3^5).
  • At x = -5, f(-5) is much larger than -g(-5) (since 0.7^{-5} is significantly larger than -1.3^{-5}).
Such comparisons elucidate the relative growths and declines of the functions across different x-values.
Comparisons using inequalities are helpful in various mathematical contexts, allowing for clear and concise analyses of relationships between functions.

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Most popular questions from this chapter

Two cities each have a population of 1.2 million people. City A is growing by a factor of 1.15 every 10 years, while city \(\mathbf{B}\) is decaying by a factor of 0.85 every 10 years. a. Write an exponential function for each city's population \(P_{A}(t)\) and \(P_{B}(t)\) after \(t\) years. b. For each city's population function generate a table of values for \(x=0\) to \(x=50,\) using 10 -year intervals, then sketch a graph of each town's population on the same grid.

On November \(25,2003,\) National Public Radio did a report on Under Armour, a sports clothing company, stating that their "profits have increased by \(1200 \%\) in the last 5 years." a. Let \(P(t)\) represent the profit of the company during every 5-year period, with \(A_{0}\) the initial amount. Write the exponential model for the company's profit. b. Assuming an initial profit of \(\$ 100,000,\) what would be the profit in year 5 ? Year \(10 ?\) c. Determine the \(a n n u a l\) growth rate for Under Armour.

The U.S. Department of Agriculture's data on per capita food commodity consumption for 1980 are listed in the following table. a. Using the data in the following table, construct exponential functions for each food category. Then evaluate each function for the year \(2000 .\) Assume \(t\) is the number of years since \(1980 .\) $$ \begin{array}{lccc} \hline & \begin{array}{c} \text { Per Capita } \\ \text { Consumption } \\ \text { (pounds) in } \\ 1980 \end{array} & \begin{array}{c} \text { Yearly } \\ \text { Growth/Decay } \\ \text { Factor } \end{array} & \begin{array}{c} \text { Exponential } \\ \text { Function } \end{array} \\ \hline \text { Beef } & 72.1 & 0.994 & B(t)= \\ \text { Chicken } & 32.7 & 1.024 & C(t)= \\ \text { Pork } & 52.1 & 0.996 & P(t)= \\ \text { Fish } & 12.4 & 1.010 & F(t)= \\ \hline \end{array} $$ b. Which commodities showed exponential growth? Which showed exponential decay? c. Write a 60 -second summary about the consumption of meat, chicken, and fish from 1980 to 2000 .

Represents an exponential function. Construct that function and then identify the corresponding growth or decay rate in percentage form. $$ \begin{array}{ccccc} \hline x & 0 & 1 & 2 & 3 \\ y & 225.00 & 228.38 & 231.80 & 235.28 \\ \hline \end{array} $$

Each of two towns had a population of 12,000 in \(1990 .\) By 2000 the population of town A had increased by \(12 \%\) while the population of town B had decreased by \(12 \%\). Assume these growth and decay rates continued. a. Write two exponential population models \(A(T)\) and \(B(T)\) for towns A and \(\mathrm{B}\), respectively, where \(T\) is the number of decades since 1990 . b. Write two new exponential models \(a(t)\) and \(b(t)\) for towns A and \(\mathrm{B}\), where \(t\) is the number of years since 1990 . c. Now find \(A(2), B(2), a(20)\), and \(b(20)\) and explain what you have found.
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