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Chapter 7: Problem 14

Think of the graphs of \(f(x)=a x^{3}\) and \(g(x)=a(3)^{x},\) where \(a>0,\) and then decide whether each of the following statements is true or false. a. \(f(x)\) and \(g(x)\) have the same vertical intercept. b. \(f(x)\) intersects \(g(x)\) only once. c. As \(x \rightarrow+\infty, g(x)>f(x)\). d. As \(x \rightarrow-\infty\), both \(f(x)\) and \(g(x)\) approach 0 .

Short Answer

Expert verified
a. False, b. True, c. True, d. False

Step by step solution

01

- Identify the vertical intercepts

For any function represented as y = f(x), the vertical intercept occurs when x = 0. Calculate the vertical intercepts for both functions, f(x) = ax^3 and g(x) = a(3)^x.
02

- Determine vertical intercept of f(x)

Substitute x = 0 in f(x):\[ f(0) = a(0)^3 = 0 \text{So, } f(x)\text{ has a vertical intercept at } (0, 0) \]
03

- Determine vertical intercept of g(x)

Substitute x = 0 in g(x):\[ g(0) = a(3)^0 = a \text{So, } g(x)\text{ has a vertical intercept at } (0, a) \]
04

- Compare vertical intercepts

The vertical intercept of f(x) is (0, 0) and the vertical intercept of g(x) is (0, a). Since a > 0, the y-intercepts are different.
05

- Evaluate intersections

Set f(x) equal to g(x) and solve: \[ ax^3 = a(3)^x \text{or } x^3 = 3^x \]To find the number of solutions, consider the graph of x^3 and the graph of 3^x. They intersect typically only once around x = 1.
06

- Compare behavior as x approaches positive infinity

Analyze the limit behavior for both functions as x approaches infinity: \[ \text{For } f(x) = ax^3, \text{as } x \rightarrow +\text{infinity}, f(x) \rightarrow +\text{infinity} \]\[ \text{For } g(x) = a(3)^x, \text{as } x \rightarrow +\text{infinity}, g(x) \rightarrow +\text{infinity} \text{ at a much faster rate} \]Thus, as x approaches positive infinity, g(x) grows faster than f(x).
07

- Compare behavior as x approaches negative infinity

Analyze the limit behavior for both functions as x approaches negative infinity: \[ \text{For } f(x) = ax^3, \text{as } x \rightarrow -\text{infinity}, f(x) \rightarrow -\text{infinity} \]\[ \text{For } g(x) = a(3)^x, \text{as } x \rightarrow -\text{infinity}, g(x)\text{ approaches 0 because } (3)^x\rightarrow 0 \text{as } x \rightarrow -\text{infinity} \]Thus, as x approaches negative infinity, f(x) does not approach 0, but g(x) does.
08

- Conclusion

Summarize whether each statement is true or false: a. False b. True c. True d. False

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Key Concepts

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

Vertical Intercept
The vertical intercept of a function is where its graph crosses the y-axis. To find the vertical intercept, you set \(x = 0\) in the function's equation. For the functions \(f(x) = ax^3\) and \(g(x) = a(3)^x\), let's see where they intercept the y-axis:
For \(f(x)\): When \(x = 0\), \( f(0) = a(0)^3 = 0\). Hence, the vertical intercept is at the point \((0, 0)\).
For \(g(x)\): When \(x = 0\), \( g(0) = a(3)^0 = a\). Thus, the vertical intercept is at the point \((0, a)\).
Since \(a > 0\), \(0 eq a\). Hence, the vertical intercepts are different between \(f(x)\) and \(g(x)\).
This means that statement (a) is false.
Function Intersection
The intersection point(s) of two functions occur where their values are equal for the same \(x\). To find this for \(f(x)\) and \(g(x)\), set them equal:
\[ax^3 = a(3)^x \text{ or } x^3 = 3^x\]
We need to examine where these two graphs might intersect, which can often be visualized or calculated. Typically, the functions \(x^3\) and \(3^x\) intersect only once around \(x = 1\). Here's how:
  • Graphing both functions can show they cross only once.
  • Otherwise, algebraic or numerical methods often indicate a unique solution.

Therefore, statement (b) is true.
Behavior at Infinity
To understand how functions behave as \(x\) goes to \(+\infty\) (positive infinity) or \(-\infty\) (negative infinity), we analyze the limits:
  • For \(f(x) = ax^3\): As \(x \to +\infty\), \(x^3 \to +\infty\). Hence, \(f(x) \to +\infty\).
  • For \(g(x) = a(3)^x\): As \(x \to +\infty\), \((3)^x \to +\infty\). But \(3^x\) grows much faster than \(x^3\).
Consequently, \(g(x)\) increases much faster than \(f(x)\) as \(x \rightarrow +\infty\). This means statement (c) is true.
For behavior as \(x \to -\infty\):
  • \(f(x) = ax^3\): As \(x \to -\infty\), \(x^3\) also goes to \(-\infty\). Hence, \(f(x)\) goes to \(-\infty\).
  • \(g(x) = a(3)^x\): As \(x \to -\infty\), \((3)^x \rightarrow 0\).
Thus, as \(x \rightarrow -\infty\), \(g(x)\) approaches 0 but \(f(x)\) does not. Therefore, statement (d) is false.
Exponential Growth
Exponential growth describes when quantities increase at a consistent rate over equal increments. This is a key feature of functions in the form \(a b^x\):

  • For \(g(x) = a(3)^x\), the term \((3)^x\) rapidly increases as \(x\) becomes larger due to the base (3) being greater than 1.
  • This results in extremely fast growth compared to polynomial functions like \(f(x) = ax^3\).
The main takeaway is that exponentials grow vastly faster than polynomials or linear functions. Understanding this growth behavior helps explain why \(g(x)\) overtakes \(f(x)\) as \(x\) goes to \(+\infty\). Exponential growth is a crucial concept in various fields, including biology (e.g., population growth) and finance (e.g., compound interest).

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Most popular questions from this chapter

a. Construct an equation to represent a relationship where w is directly proportional to both \(y\) and \(z\) and inversely proportional to the square of \(x\). b. Assume that \(w=10\) when \(y=12, z=15,\) and \(x=6 .\) Find \(k,\) the constant of proportionality. c. Using your equation from part (b), find \(x\) when \(w=2\), \(y=5,\) and \(z=6\)

Write a general formula to describe each variation. Use the information given to find the constant of proportionality. a. \(Q\) is directly proportional to both the cube root of \(t\) and the square of \(d\), and \(Q=18\) when \(t=8\) and \(d=3\). b. \(A\) is directly proportional to both \(h\) and the square of the radius, \(r\), and \(A=100 \pi\) when \(r=5\) and \(h=2\). c. \(V\) is directly proportional to \(B\) and \(h,\) and \(V=192\) when \(B=48\) and \(h=4\) d. \(T\) is directly proportional to both the square root of \(p\) and the square of \(u,\) and \(T=18\) when \(p=4\) and \(u=6\).

The radius of Earth is about 6400 kilometers. Assume that Earth is spherical. Express your answers to the questions below in scientific notation. a. Find the surface area of Earth in square meters. b. Find the volume of Earth in cubic meters. c. Find the ratio of the surface area to the volume.

When installing Christmas lights on the outside of your house, you read the warning "Do not string more than four sets of lights together." This is because the electrical resistance, \(R,\) of wire varies directly with the length of the wire, \(l,\) and inversely with the square of the diameter of the wire, \(d\). a. Construct an equation for electrical wire resistance. b. If you double the wire diameter, what happens to the resistance? c. If you increase the length by \(25 \%\) (say, going from four to five strings of lights), what happens to the resistance?

Consider two solid figures, a sphere and cylinder, where each has radius \(r\). The volume of a sphere is \(V_{s}=\frac{4}{3} \pi r^{3}\) and the volume of a cylinder is \(V_{c}=\pi r^{2} h\). a. If \(h=1\), when is the volume of the sphere greater than the volume of the cylinder? b. What value of \(h\) would make the volumes the same? c. Complete this statement: "The volume of the cylinder with radius \(r\) is greater than the volume of a sphere of radius \(r,\) when \(h>\) ______ ;`
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