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Chapter 5: Problem 8

Generate quick sketches of each of the following functions, without the aid of technology. $$ \begin{array}{ll} f(x)=4(3.5)^{x} & g(x)=4(0.6)^{x} \\ h(x)=4+3 x & k(x)=4-6 x \end{array} $$ a. As \(x \rightarrow+\infty\), which function(s) approach \(+\infty ?\) b. As \(x \rightarrow+\infty\), which function(s) approach \(0 ?\) c. As \(x \rightarrow-\infty\), which function(s) approach \(-\infty\) ? d. As \(x \rightarrow-\infty\), which function(s) approach 0 ?

Short Answer

Expert verified
a. \(f(x)\), \(h(x)\); b. \(g(x)\); c. \(h(x)\), \(k(x)\); d. \(f(x)\), \(g(x)\)

Step by step solution

01

Identify Functions Behavior as x -> +鈭�

Analyze the functions as they approach positive infinity. For exponential growth functions like \(f(x) = 4(3.5)^x\), the function will approach \(+\infty\) since the base of the exponent is greater than 1. For linear functions with a positive slope, such as \(h(x) = 4 + 3x\), these also approach \(+\infty\).
02

Identify Functions That Approach 0 as x -> +鈭�

Consider the functions as they approach zero. The function \(g(x) = 4(0.6)^x\) has a base less than 1. As \(x\) increases, \((0.6)^x\) gets closer to zero, so \(g(x)\) approaches zero.
03

Identify Functions Behavior as x -> -鈭�

For linear functions \(h(x) = 4 + 3x\) and \(k(x) = 4 - 6x\), analyze their slopes. As \(x\) approaches \(-\infty\), linear functions with a positive slope will approach \(-\infty\). Therefore, \(k(x)\) approaches \(-\infty\).
04

Identify Functions That Approach 0 as x -> -鈭�

Look at the exponential functions \(f(x) = 4(3.5)^x\) and \(g(x) = 4(0.6)^x\). As \(x\) approaches \(-\infty\), the exponential function \(f(x)\) with a base greater than 1 approaches zero. Similarly, \(g(x)\) will also approach \(0\) because \(0.6\) raised to a large negative power approaches zero.

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Key Concepts

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

exponential growth
Exponential growth occurs when the rate of growth in a function is proportional to the function's current size. As the input value increases, the function grows at an increasingly rapid rate. A classic example is the function \(f(x)=4(3.5)^x\). Here, the base of the exponent, \(3.5\), is greater than 1, causing the function to grow exponentially as \(x\) increases. The larger the base, the faster the growth rate. Exponential growth is commonly seen in populations, investments, and certain natural phenomena where doubling behavior is observed.

In our case:
  • \(f(x)=4(3.5)^x\) represents an exponentially growing function that approaches \(+\text{infty}\) as \(x\) heads towards \(+\text{infty}\)
  • Conversely, exponential functions like \(g(x)=4(0.6)^x\), where the base is less than 1, will decay towards 0 as \(x\) increases.
linear functions
Linear functions are functions of the form \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept. The slope indicates how steep the line is and the direction in which it moves. If \(m < 0\), the function decreases as \(x\) increases, and if \(m > 0\), the function increases as \(x\) increases.

For example:
  • \(h(x) = 4 + 3x\) is a linear function with a positive slope of \(3\). As \(x\) heads towards \(+\text{infty}\), \(h(x)\) approaches \(+\text{infty}\).
  • \(k(x) = 4 - 6x\) is another linear function, but with a negative slope of \(-6\). This means \(k(x)\) will decrease towards \(-\text{infty}\) as \(x\) approaches \(-\text{infty}\).
Linear functions are straightforward to graph and analyze, making them fundamental in understanding more complex functions.
limits at infinity
Limits at infinity involve understanding the behavior of functions as \(x\) approaches \(+\text{infty}\) or \(-\text{infty}\). Recognizing these limits helps in predicting the end behavior of a function. Some functions grow without bound, while others approach specific limiting values.

In our analysis:
  • As \(x\) approaches \(+\text{infty}\), \(f(x) = 4(3.5)^x\) and \(h(x) = 4 + 3x\) approach \(+\text{infty}\).
  • However, \(g(x) = 4(0.6)^x\) approaches 0 as \(x\) increases because \(0.6^x\) shrinks towards 0.
  • As \(x\) approaches \(-\text{infty}\), \(k(x) = 4 - 6x\) approaches \(-\text{infty}\), and both \(f(x) = 4(3.5)^x\) and \(g(x) = 4(0.6)^x\) approach 0.
The concepts of limits are crucial for understanding the long-term behavior of various types of functions.
function sketching
Sketching functions is a vital skill in mathematics. It involves plotting a function based on its algebraic formula and understanding its key characteristics, such as intercepts, slopes, and asymptotic behavior. For exponential and linear functions, this process helps visualize their growth or decay.

For instance:
  • When sketching \(f(x)=4(3.5)^x\), start by plotting a few points for small values of \(x\) and capitalize on the rapid growth observed as \(x\) increases.
  • For \(g(x)=4(0.6)^x\), notice the quick decline towards zero. This function will get closer to the x-axis as \(x\) increases.
  • Linear functions like \(h(x)=4+3x\) and \(k(x)=4-6x\) can be sketched by identifying the y-intercept and plotting additional points based on the slope.

Function sketching not only aids in visualizing the behavior of functions but also forms a foundation for more intricate graphing techniques in advanced mathematics.

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Most popular questions from this chapter

[Part (e) requires use of the Internet and technology to find a best-fit function.] A "rule of thumb" used by car dealers is that the trade-in value of a car decreases by \(30 \%\) each year. a. Is this decline linear or exponential? b. Construct a function that would express the value of the car as a function of years owned. c. Suppose you purchase a car for \(\$ 15,000 .\) What would its value be after 2 years? d. Explain how many years it would take for the car in part (c) to be worth less than \(\$ 1000\). Explain how you arrived at your answer. e. Internet search: Go to the Internet site for the Kelley Blue Book (www.kbb.com). i. Enter the information about your current car or a car you would like to own. Specify the actual age and mileage of the car. What is the Blue Book value? ii. Keeping everything else the same, assume the car is I year older and increase the mileage by 10,000 . What is the new value? iii. Find a best-fit exponential function to model the value of your car as a function of years owned. What is the annual decay rate? iv. According to this function, what will the value of your car be 5 years from now?

Determine which of the following functions are exponential. Identify each exponential function as representing growth or decay and find the vertical intercept. a. \(A=100\left(1.02^{t}\right)\) b. \(f(x)=4\left(3^{x}\right)\) c. \(g(x)=0.3\left(10^{x}\right)\) d. \(y=100 x+3\) e. \(M=2^{p}\) f. \(y=x^{2}\)

Write an equation for an exponential decay function where: a. The initial population is 10,000 and the decay factor is \(\frac{2}{3}\). b. The initial population is \(2.7 \cdot 10^{13}\) and the decay factor is 0.27 c. The initial population is 219 and the population drops to one-tenth its previous size during each time period.

(Graphing program recommended.) A small village has an initial size of 50 people at time \(t=0,\) with \(t\) in years. a. If the population increases by 5 people per year, find the formula for the population size \(P(t)\). b. If the population increases by a factor of 1.05 per year, find a new formula \(Q(t)\) for the population size. c. Plot both functions on the same graph over a 30 -year period. d. Estimate the coordinates of the point(s) where the graphs intersect. Interpret the meaning of the intersection point(s).

The price of a home in Medford was \(\$ 100,000\) in 1985 and rose to \(\$ 200,000\) in 2005 . a. Create two models, \(f(t)\) assuming linear growth and \(g(t)\) assuming exponential growth, where \(t=\) number of years after \(1985 .\) b. Fill in the following table representing linear growth and exponential growth for \(t\) years after \(1985 .\) c. Which model do you think is more realistic?
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