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Chapter 7: Problem 10

Given the functions \(f(x)=x^{4}\) and \(g(x)=(4)^{x}\) a. Find \(f(3 x)\) and \(3 f(x)\). Summarize the difference between these functions and \(f(x)\). b. Find \(g(3 x)\) and \(3 g(x)\). Summarize the difference between these functions and \(g(x)\).

Short Answer

Expert verified
For both functions, multiplying the input scales the function's growth more significantly than multiplying the output directly.

Step by step solution

01

- Find \( f(3x) \)

Given the function \( f(x) = x^4 \), substitute \( 3x \) for \( x \). \[ f(3x) = (3x)^4 = 81x^4 \]
02

- Find \( 3f(x) \)

Multiply the function \( f(x) = x^4 \) by 3. \[ 3f(x) = 3 \times x^4 = 3x^4 \]
03

- Summarize differences for \( f(3x) \) and \( 3f(x) \)

\( f(3x) \) scales the input of the function and then applies the function, resulting in a larger growth rate, whereas \( 3f(x) \) scales the output after applying the original function, resulting in a linear multiplication.
04

- Find \( g(3x) \)

Given the function \( g(x) = 4^x \), substitute \( 3x \) for \( x \). \[ g(3x) = 4^{3x} = 4^{3x} \]
05

- Find \( 3g(x) \)

Multiply the function \( g(x) = 4^x \) by 3. \[ 3g(x) = 3 \times 4^x = 3 \times 4^x \]
06

- Summarize differences for \( g(3x) \) and \( 3g(x) \)

\( g(3x) \) scales the input of the function, resulting in exponential growth due to the power being multiplied by 3, whereas \( 3g(x) \) scales the output of the function by a factor of 3.

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Key Concepts

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

input scaling
In mathematics, input scaling changes how a function behaves by modifying its input value. When you see something like \( f(3x) \), this means the input \( x \) is scaled by the factor of 3 before any calculation is made. Let's break it down with our example function, \( f(x) = x^4 \). If we scale the input by 3, then we get \( f(3x) = (3x)^4 \). Simplifying this, we find \( (3x)^4 = 81x^4 \). As you can see, input scaling changes the rate at which the function grows. Instead of just raising \( x \) to the fourth power, scaling the input by 3 makes the expression much larger. This means the function grows at a much faster rate.
output scaling
Output scaling modifies the result of the function after the initial calculation is performed. For example, with our function \( f(x) = x^4 \), if we want to scale the output by 3, we would compute \( 3f(x) \). This means we first find \( f(x) = x^4 \) and then multiply the result by 3. So, \( 3f(x) = 3x^4 \). Unlike input scaling, output scaling does not change the function's growth rate before the calculation. Instead, it simply multiplies the result. It's much like increasing the amplitude or height of the function's output.
exponential growth
Exponential growth refers to an increase in quantity over time that follows an exponential function, which means the growth rate becomes ever more rapid. For instance, look at \( g(x) = 4^x \). If we scale the input by 3, the function transforms into \( g(3x) = 4^{3x} \). This means the input is multiplied by 3, causing the function's value to explode much faster than normal. On the other hand, if we scale the output, we would multiply by 3 after finding \( g(x) = 4^x \). Thus, \( 3g(x) = 3 \times 4^x \). While this also makes the function value larger, it does not boost the rate of exponential growth as input scaling does. Instead, it only increases the resulting value.
polynomial functions
Polynomial functions are algebraic expressions consisting of variables raised to whole-number exponents and their coefficients. Commonly, they are represented like \( f(x) = x^4 \). When dealing with polynomial functions, the concepts of input and output scaling affect how these functions grow and behave. With input scaling, such as \( f(3x) \), we see a more significant and rapid increase in the function's output than when just scaling the output, like \( 3f(x) \). Understanding how scaling impacts polynomial functions can help us predict and analyze their behavior in different scenarios. This foundational knowledge is critical for tackling more complex mathematical problems.

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Most popular questions from this chapter

(Graphing program recommended.) Assume a person weighing \(P\) pounds is standing at the center of a \(4^{\prime \prime} \times 12^{\prime \prime}\) fir plank that spans a distance of \(L\) feet. The downward deflection \(D_{\text {deflection }}\) (in inches) of the plank can be described by $$ D_{\text {deflection }}=\left(5.25 \cdot 10^{-7}\right) \cdot P \cdot L^{3} $$ a. Graph the deflection formula, \(D_{\text {deflection }}\), assuming \(P=200\) pounds. Put values of \(L\) (from 0 to 25 feet) on the horizontal axis and deflection (in inches) on the vertical axis. b. A rule used by architects for estimating acceptable deflection, \(D_{\text {safe }},\) in inches, of a beam \(L\) feet long bent downward as a result of carrying a load is $$ D_{\text {safe }}=0.05 L $$ Add to your graph from part (a) a plot of \(D_{\text {safe. }}\) c. Is it safe for a 200 -pound person to sit in the middle of a \(4^{\prime \prime} \times 12^{\prime \prime}\) fir plank that spans 20 feet? What maximum span is safe for a 200 -pound person on a \(4^{\prime \prime} \times 12^{\prime \prime}\) plank?

(Graphing program optional.) Plot the functions \(f(x)=x^{2}\), \(g(x)=3 x^{2},\) and \(h(x)=\frac{1}{4} x^{2}\) on the same grid. Insert the symbol \(>\) or \(<\) to make the relation true. a. For \(x>0, g(x) \quad f(x) __________ h(x)\) b. For \(x<0, g(x) \quad f(x) __________ h(x)\)

Suppose you are traveling in your car at speed \(S\) and you suddenly brake hard, leaving skid marks on the road. A "rule of thumb" for the distance, \(D,\) that the car will skid is given by $$ D=\frac{S^{2}}{30 f} $$ where \(D=\) distance the car skids (in feet) \(, S=\) speed of the car (in miles per hour), and \(f\) is a number called the coefficient of friction that depends on the road surface and its condition. For a dry tar road, \(f \approx 1.0 .\) For a wet tar road, \(f \approx 0.5 .\) (We saw a variation of this problem in Example 7 , Section \(7.2 .\) ) a. What is the equation giving distance skidded as a function of speed for a dry tar road? For a wet tar road? b. Generate a small table of values for both functions in part (a), including speeds between 0 and 100 miles per hour. c. Plot both functions on the same grid. d. Why do you think the coefficient of friction is less for a wet road than for a dry road? What effect does this have on the graph in part (c)? e. In the accompanying table, estimate the speed given the distances skidded on dry and on wet tar roads. Describe the method you used to find these numbers. f. If one car is going twice as fast as another when they both jam on the brakes, how much farther will the faster car skid? Explain. Does your answer depend on whether the road is dry or wet? $$ \begin{array}{cl} \hline \text { Distance Skidded } &{\text { Estimated Speed (mph) }} \\ \ { 2 - 3 } \text { (ft) } & \text { Dry Tar } & \text { Wet Tar } \\ \hline 25 & & \\ 50 & & \\ 100 & & \\ 200 & & \\ 300 & & \\ \hline \end{array} $$

Sketch by hand the graph of each function: \(f(x)=x^{3}, \quad g(x)=-x^{3}, \quad h(x)=\frac{1}{2} x^{3}, \quad j(x)=-2 x^{3}\) a. Identify the \(k\) value, the constant of proportionality, for each function. b. Which graph is a reflection of \(f(x)\) across the \(x\) -axis? c. Which graph is both a stretch and a reflection of \(f(x)\) across the \(x\) -axis? d. Which graph is a compression of \(f(x)\) ?

The distance, \(d,\) a ball travels down an inclined plane is directly proportional to the square of the total time, \(t\), of the motion. a. Express this relationship as a function where \(d\) is the dependent variable. b. If a ball starting at rest travels a total of 4 feet in 0.5 second, find the value of the constant of proportionality in part (a). c. Complete the equation and solve for \(t\). Is \(t\) directly proportional to \(d ?\)
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