Chapter 4

In Exercises 1-4 is the first quantity proportional to the second quantity? If so, what is the constant of proportionality? \(d\) is the distance traveled in miles and \(t\) is the time traveled in hours at a speed of \(50 \mathrm{mph}\).

Find the average rate of change of \(f(x)=x^{2}+3 x\) on the intervals indicated. Between 2 and 4 .

Solve \(f(x)=0\) for \(x\). $$ f(x)=\sqrt{x-2}-4 $$

In Exercises \(1-4\) (a) Evaluate the function at the given input values. Which gives the greater output value? (b) Explain the answer to part (a) in terms of the algebraic expression for the function. $$ f(x)=9-x, x=1,3 $$

In Exercises 1-2, write the relationship using function notation (that is, \(y\) is a function of \(x\) is written \(y=f(x)\) ). Weight, \(w\), is a function of caloric intake, \(c\).

Find the average rate of change of \(f(x)=x^{2}+3 x\) on the intervals indicated. Between -2 and 4 .

Solve \(f(x)=0\) for \(x\). $$ f(x)=6-3 x $$

(a) Evaluate the function at the given input values. Which gives the greater output value? (b) Explain the answer to part (a) in terms of the algebraic expression for the function. $$ g(a)=a-2, a=-5,-2 $$

Write the relationship using function notation (that is, \(y\) is a function of \(x\) is written \(y=f(x)\) ). Number of molecules, \(m\), in a gas, is a function of the volume of the gas, \(v\)

Is the first quantity proportional to the second quantity? If so, what is the constant of proportionality? \(p\) is the sale price of an item whose original price is \(p_{0}\) in a \(30 \%\) off sale.

Find the average rate of change of \(f(x)=x^{2}+3 x\) on the intervals indicated. Between -4 and -2 .

Solve \(f(x)=0\) for \(x\). $$ f(x)=4 x-9 $$

(a) Evaluate the function at the given input values. Which gives the greater output value? (b) Explain the answer to part (a) in terms of the algebraic expression for the function. $$ C(p)=\frac{-p}{5}, p=100,200 $$

The number, \(N,\) of napkins used in a restaurant is \(N=f(C)=2 C,\) where \(C\) is the number of customers. What is the dependent variable? The independent variable?

Find the average rate of change of \(f(x)=x^{2}+3 x\) on the intervals indicated. Between 3 and 1 .

Solve \(f(x)=0\) for \(x\). $$ f(x)=2 x^{2}-18 $$

(a) Evaluate the function at the given input values. Which gives the greater output value? (b) Explain the answer to part (a) in terms of the algebraic expression for the function. $$ h(t)=\frac{t}{5}, t=4,6 $$

A silver mine's profit, \(P,\) is \(P=g(s)=-100,000+\) \(50,000 s\) dollars, where \(s\) is the price per ounce of silver. What is the dependent variable? The independent variable?

Find the average rate of change of \(g(x)=2 x^{3}-3 x^{2}\) on the intervals indicated. Between 1 and 3 .

Solve \(f(x)=0\) for \(x\). $$ f(x)=2 \sqrt{x}-10 $$

In Exercises \(5-8, f(t)=t / 2+7\). Determine whether the two expressions are equivalent. $$ \frac{f(t)}{3}, \frac{1}{3} f(t) $$

In Exercises 5-6, evaluate the function for \(x=-7\). $$ f(x)=x / 2-1 $$

For each of the formulas in Exercises 5-13, is \(y\) directly proportional to \(x ?\) If so, give the constant of proportionality. $$ y=x \cdot 7 $$

Find the average rate of change of \(g(x)=2 x^{3}-3 x^{2}\) on the intervals indicated. Between -1 and 4 . .

Solve \(f(x)=0\) for \(x\). $$ f(x)=2(2 x-3)+2 $$

Find the average rate of change of \(g(x)=2 x^{3}-3 x^{2}\) on the intervals indicated. Between 0 and 10 .

Solve the equation \(g(t)=a\) given that: $$ a(t)=6-t \text { and } a=1 $$

Let \(g(x)=(12-x)^{2}-(x-1)^{3}\). Evaluate (a) \(g(2)\) (b) \(g(5)\) (c) \(g(0)\) (d) \(g(-1)\)

For each of the formulas in Exercises 5-13, is \(y\) directly proportional to \(x ?\) If so, give the constant of proportionality. $$ y=\sqrt{5} \cdot x $$

Find the average rate of change of \(g(x)=2 x^{3}-3 x^{2}\) on the intervals indicated. Between -0.1 and 0.1 .

Solve the equation \(g(t)=a\) given that: $$ g(t)=(2 / 3) t+6 \text { and } a=10 $$

Let \(f(x)=2 x^{2}+7 x+5\). Evaluate (a) \(f(3)\) (b) \(f(a)\) (c) \(f(2 a)\) (d) \(f(-2)\)

For each of the formulas in Exercises 5-13, is \(y\) directly proportional to \(x ?\) If so, give the constant of proportionality. $$ y=x / 9 $$

The value in dollars of an investment \(t\) years after 2003 is given by $$V=1000 \cdot 2^{t / 6}$$ Find the average rate of change of the investment's value between 2004 and 2007 .

In Exercises \(9-16\), are the two functions the same function? $$ f(x)=x^{2}-4 x+5 \text { and } g(x)=(x-2)^{2}+1 $$

Solve the equation \(g(t)=a\) given that: $$ g(t)=3(2 t-1) \text { and } a=-3 $$

In Exercises 9-14, evaluate the expressions given that $$ f(x)=\frac{2 x+1}{3-5 x} \quad g(y)=\frac{1}{\sqrt{y^{2}+1}} $$ $$ f(0) $$

Atmospheric levels of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) have risen from 336 parts per million (ppm) in 1979 to 382 parts per million (ppm) in \(2007 .^{1}\) Find the average rate of change of \(\mathrm{CO}_{2}\) levels during this time period.

Are the two functions the same function? $$ f(x)=2(x+1)(x-3) \text { and } g(x)=x^{2}-2 x-3 $$

Solve the equation \(g(t)=a\) given that: $$ g(t)=\frac{t-1}{3} \text { and } a=1 $$

Evaluate the expressions given that $$ f(x)=\frac{2 x+1}{3-5 x} \quad g(y)=\frac{1}{\sqrt{y^{2}+1}} $$ $$ g(0) $$

Sea levels were most recently at a low point about 22,000 years ago. \({ }^{2}\) Since then they have risen approximately 130 meters. Find the average rate of change of the sea level during this time period.

Are the two functions the same function? $$ f(t)=450+30 t, \text { and } g(p)=450+30 p $$

Solve the equation \(g(t)=a\) given that: $$ g(t)=2(t-1)+4(2 t+3) \text { and } a=0 $$

Evaluate the expressions given that $$ f(x)=\frac{2 x+1}{3-5 x} \quad g(y)=\frac{1}{\sqrt{y^{2}+1}} $$ $$ g(-1) $$

Global temperatures may increase by up to \(10^{\circ} \mathrm{F}\) between 1990 and \(2100 .{ }^{3}\) Find the average rate of change of global temperatures between 1990 and 2100 .

Are the two functions the same function? $$ A(n)=(n-1) / 2 \text { and } B(n)=0.5 n-0.5 $$

Evaluate the expressions given that $$ f(x)=\frac{2 x+1}{3-5 x} \quad g(y)=\frac{1}{\sqrt{y^{2}+1}} $$ $$ f(10) $$

Table 4.14 gives values of \(D=f(t),\) the total US debt (in \$ billions) \(t\) years after \(2000 .{ }^{4}\) Answer based on this information. $$\begin{aligned}&\text { Table }\\\ &4.14\\\&\begin{array}{c|r}\hline t & D \text { (\$ billions) } \\\\\hline 0 & 5674.2 \\\1 & 5807.5 \\\2 & 6228.2 \\\3 & 6783.2 \\\4 & 7379.1 \\\5 & 7932.7 \\\6 & 8507.0 \\\7 & 9007.7 \\\8 & 10,024.7 \\ \hline\end{array}\end{aligned}$$ 13\. Evaluate $$\frac{f(5)-f(1)}{5-1}$$and say what it tells you about the US debt.

Are the two functions the same function? $$ r(x)=5(x-2)+3 \text { and } s(x)=5 x+7 $$