Chapter 2

Find the open interval in which \(x\) must lie in order for the given condition to hold. \(y=5 x+12,\) and the difference between \(y\) and 4 is less than 0.0001.

Consider the function h as defined. Find functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x) .\) (There are several possible ways to do this.) $$h(x)=(2 x-3)^{3}$$

Solve each equation or inequality graphically. $$|2 x+7|=6 x-1$$

Consider the function h as defined. Find functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x) .\) (There are several possible ways to do this.) $$h(x)=\sqrt{6 x}+12$$

Solve each equation or inequality graphically. $$-|3 x-12| \geq-x-1$$

Consider the function h as defined. Find functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x) .\) (There are several possible ways to do this.) $$h(x)=\sqrt[3]{2 x+3}-4$$

Solve each equation or inequality graphically. $$|x-4|>0.5 x-6$$

For each situation, if \(x\) represents the number of items produced, (a) write a cost function, (b) find a revenue function if each item sells for the price given, (c) state the profit function, (d) determine analytically how many items must be produced before a profit is realized (assume whole numbers of items), and (e) support the result of part (d) graphically. The fixed cost is 500 dollars, the cost to produce an item is 10 dollars, and the selling price of the item is 35 dollars.

Concept Check Suppose that the graph of \(y=f(x)\) is symmetric with respect to the \(y\) -axis and is reflected across the \(y\) -axis. How will the new graph compare with the original one?

Solve each equation or inequality graphically. $$2 x+8>-|3 x+4|$$

For each situation, if \(x\) represents the number of items produced, (a) write a cost function, (b) find a revenue function if each item sells for the price given, (c) state the profit function, (d) determine analytically how many items must be produced before a profit is realized (assume whole numbers of items), and (e) support the result of part (d) graphically. The fixed cost is 180 dollars, the cost to produce an item is 11 dollars, and the selling price of the item is 20 dollars.

Solve each equation or inequality graphically. $$|3 x+4|<-3 x-14$$

For each situation, if \(x\) represents the number of items produced, (a) write a cost function, (b) find a revenue function if each item sells for the price given, (c) state the profit function, (d) determine analytically how many items must be produced before a profit is realized (assume whole numbers of items), and (e) support the result of part (d) graphically. The fixed cost is 2700 dollars, the cost to produce an item is 100 dollars, and the selling price of the item is 280 dollars.

Solve each equation or inequality graphically. $$|x-\sqrt{13}|+\sqrt{6} \leq-x-\sqrt{10}$$

For each situation, if \(x\) represents the number of items produced, (a) write a cost function, (b) find a revenue function if each item sells for the price given, (c) state the profit function, (d) determine analytically how many items must be produced before a profit is realized (assume whole numbers of items), and (e) support the result of part (d) graphically. The fixed cost is 1000 dollars, the cost to produce an item is 200 dollars, and the selling price of the item is 240 dollars.

The formula for the volume of a sphere is \(V=\frac{4}{3} \pi r^{3},\) where \(r\) represents the radius of the sphere. (a) Write a function \(D(r)\) that gives the volume gained when the radius of a sphere of \(r\) inches is increased by 3 inches. (b) Graph \(y=D(r)\) found in part (a), using \(x\) for \(r,\) in the window \([0,10]\) by \([0,1500]\) (c) Use your calculator to graphically find the amount of volume gained when a sphere with a 4-inch radius is increased to a 7 -inch radius. (d) Verify your result in part (c) analytically.

Suppose that the length of a rectangle is twice its width. Let \(x\) represent the width of the rectangle. (a) Write a formula for the perimeter \(P\) of the rectangle in terms of \(x\) alone. Then use \(P(x)\) notation to describe it as a function. What type of function is this? (b) Graph the function \(P\) found in part (a) in the window \([0,10]\) by \([0,100] .\) Locate the point for which \(x=4,\) and explain what \(x\) represents and what \(y\) represents. (c) On the graph of \(P\), locate the point with \(x\) -value 4 Then sketch a rectangle satisfying the conditions described earlier, and evaluate its perimeter if its width is this \(x\) -value. Use the standard perimeter formula. How does the result compare with the \(y\) -value shown on your screen? (d) On the graph of \(P\), find a point with an integer \(y\) -value. Interpret the \(x\) - and \(y\) -coordinates here.

The perimeter \(x\) of a square with side length \(s\) is given by the formula $$x=4 s$$ (a) Solve for \(s\) in terms of \(x\) (b) If \(y\) represents the area of this square, write \(y\) as a function of the perimeter \(x\) (c) Use the composite function of part (b) to analytically find the area of a square with perimeter \(6 .\) (d) Support the result of part (c) graphically, and explain the result.

The area \(\mathscr{A}\) of an equilateral triangle with sides of length \(x\) is given by $$\mathscr{A}(x)=\frac{\sqrt{3}}{4} x^{2}$$ (a) Find \(\mathscr{A}(2 x),\) the function representing the area of an equilateral triangle with sides of length twice the original length. (b) Find analytically the area of an equilateral triangle with side length 16. Use the given formula for \(\mathscr{A}(x)\) (c) Support the result of part (b) graphically.

When a thermal inversion layer is over a city, pollutants cannot rise vertically, but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 A.M. and that the pollutant disperses horizontally over a circular area. Let \(t\) represent the time in hours since the factory began emitting pollutants \((t=0\) represents 8 A.M.) and assume that the radius of the circle of pollution is $$r(t)=2 t \quad \text{miles}.$$ The area of a circle of radius \(r\) is represented by $$\mathscr{A}(r)=\pi r^{2}$$ (a) Find \((\mathscr{A} \circ r)(t)\) (b) Interpret \((\mathscr{A} \circ r)(t)\) (c) What is the area of the circular region covered by the layer at noon? (d) Support your result graphically.

The following table lists the (projected) average age \(A\) for a person living during year \(x,\) and also the combined total of years \(T\) in billions lived by the current world population during year \(x\). $$\begin{array}{c|c|c|c|c}\boldsymbol{x} & 1950 & 2000 & 2050 & 2100 \\\\\hline \boldsymbol{A}(\boldsymbol{x}) & 28 & 30 & 38 & 42 \\\\\hline \boldsymbol{T}(\boldsymbol{x}) & 80 & 180 & 360 & 430\end{array}$$ (a) Evaluate \(A(2100)\) and \(T(2100)\). Interpret your results. (b) Evaluate \(\frac{T(2100)}{A(2100)}\). Interpret your result. (c) Let \(P(x)=\frac{T(x)}{A(x)} .\) Interpret what \(P(x)\) calculates.

The table below shows the acreage, in millions, of the total of corn and soybeans harvested annually in the United States. In the table, \(x\) represents the year and \(f\) computes the total number of acres for these two crops. The function \(g\) computes the number of acres for corn only. $$\begin{array}{c|c|c|c|c}\boldsymbol{x} & 2009 & 2010 & 2011 & 2012 \\\\\hline \boldsymbol{f}(\boldsymbol{x}) & 164.0 & 166.3 & 167.6 & 172.5 \\\\\hline \boldsymbol{g}(\boldsymbol{x}) & 86.5 & 88.2 &92.3 & 96.4\end{array}$$ (a) Make a table for a function \(h\) that is defined by the equation \(h(x)=f(x)-g(x)\) (b) Interpret what \(h\) computes.

A common air pollutant responsible for acid rain is sulfur dioxide \(\left(\mathrm{SO}_{2}\right) .\) Emissions of \(\mathrm{SO}_{2}\) during year \(x\) are computed by \(f(x)\) in the table. Emissions of carbon monoxide (CO) are computed by \(g(x)\) Amounts are given in millions of tons. $$\begin{array}{c|c|c|c|c|c}\boldsymbol{x} & 1970 & 1980 & 1990 & 2000 & 2010 \\\\\hline \boldsymbol{f}(\boldsymbol{x}) & 31.2 & 25.9 & 23.1 & 16.3 & 13.0 \\\\\hline \boldsymbol{g}(\boldsymbol{x}) & 204.0 &185.4 & 154.2 & 114.5 & 74.3\end{array}$$ (a) Evaluate \((f+g)(2010)\) (b) Interpret \((f+g)(x)\) (c) Make a table for \((f+g)(x)\)

The greenhouse gas methane lets sunlight into the atmosphere, but blocks heat from escaping the earth's atmosphere. Methane is a by-product of burning fossil fuels. In the table, \(f\) models the predicted methane emissions in millions of tons produced by developed countries during year \(x .\) The function \(g\) models the same emissions for developing countries. $$\begin{array}{c|c|c|c|c|c}\boldsymbol{x} & 1990 & 2000 & 2010 & 2020 & 2030 \\\\\hline \boldsymbol{f}(\boldsymbol{x}) & 27 & 28 & 29 & 30 & 31 \\\\\hline \boldsymbol{g}(\boldsymbol{x}) & 5 & 7.5 & 10 & 12.5 & 15\end{array}$$ (a) Make a table for a function \(h\) that models the total predicted methane emissions for developed and developing countries. (b) Write an equation that relates \(f(x), g(x),\) and \(h(x)\)

During the early years of the AIDS epidemic, cases and cumulative deaths reported for selected years \(x\) could be modeled by quadratic functions. For \(1982-\) 1994 , the numbers of AIDS cases are modeled by $$f(x)=3200(x-1982)^{2}+1586$$ and the numbers of deaths are modeled by $$g(x)=1900(x-1982)^{2}+619$$ $$\begin{array}{|l|c|c|}\hline \text { Year } & \text { Cases } & \text { Deaths } \\\\\hline 1982 & 1,586 & 619 \\ 1984 & 10,927 & 5,605 \\\1986 & 41,910 & 24,593 \\\1988 & 106,304 & 61,911 \\\1990 & 196,576 & 120,811 \\ 1992 & 329,205 & 196,283 \\\1994 & 441,528 & 270,533\end{array}$$ (a) Graph \(h(x)=\frac{g(x)}{f(x)}\) in the window \([1982,1994]\) by \([0,1] .\) Interpret the graph. (b) Compute the ratio \(\frac{\text { deaths }}{\text { cases }}\) for each year. Compare the results with those from part (a).