Chapter 2

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=-6$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=\frac{1}{2 x}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|4 x+1|=|4 x+6|$$

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=|-x|$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=\frac{1}{x^{2}}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|6 x+9|=|6 x-3|$$

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=\frac{1}{4 x^{3}}$$

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.) $$h(x)=(6 x-2)^{2}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|0.25 x+1|=|0.75 x-3|$$

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=\sqrt{x^{2}}$$

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.) $$h(x)=\left(11 x^{2}+12 x\right)^{2}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|0.40 x+2|=|0.60 x-5|$$

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.) $$h(x)=\sqrt{x^{2}-1}$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|3 x+10|=|-3 x-10|$$

An equation of the form \(|f(x)|=|g(x)|\) is given. (a) Solve the equation analytically and support the solution graphically. (b) Solve \(|f(x)|>|g(x)|\). (c) Solve \(|f(x)|<|g(x)|\). $$|5 x-6|=|-5 x+6|$$

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 graphically. $$|x+1|+|x-6|=11$$

Solve each equation graphically. $$|2 x+2|+|x+1|=9$$

Solve the following. Complete the following. (a) Write a function \(F\) that converts \(x\) miles to feet. (b) Write a function \(I\) that converts \(x\) feet to inches. (c) Write a formula for the composition \((I \circ F)(x)\) (d) Explain what \((I \circ F)(x)\) calculates.

Solve each equation graphically. $$|x|+|x-4|=8$$

show the graphs of \(Y_{1}=\sqrt[3]{X}\) and \(Y_{2}=5 \sqrt[3]{X}\). The point whose coordinates are given at the bottom of the screen lies on the graph of \(\mathrm{Y}_{1} .\) Use this graph, not your calculator, to find the coordinates of the corresponding point on the graph of \(\mathrm{Y}_{2}\) for the same value of \(\mathrm{X}\) shown.

Solve the following. Complete the following. (a) Write a function \(T\) that converts \(x\) tons to pounds. (b) Write a function \(O\) that converts \(x\) pounds to ounces. (c) Write a formula for the composition \((O \circ T)(x)\) (d) Explain what \((O \circ T)(x)\) calculates.

Solve each equation graphically. $$|0.5 x+2|+|0.25 x+4|=9$$

For each simation, 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 analyrically how many ilems 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 \(\$ 500\), the cost to produce an item is \(\$ 10\), and the selling price of the item is \(\$ 35\).

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-50| \leq 22,\) Boston, Massachusetts

For each simation, 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 analyrically how many ilems 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 \(\$ 180\), the cost to produce an item is \(\$ 11\), and the selling price of the item is \(\$ 20\).

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-10| \leq 36,\) Chesterfield, Canada

For each simation, 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 analyrically how many ilems 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 \(\$ 2700\), the cost to produce an item is \(\$ 100,\) and the selling price of the item is \(\$ 280\)

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-61.5| \leq 12.5,\) Buenos Aires, Argentina

For each simation, 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 analyrically how many ilems 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 \(\$ 1000,\) the cost to produce an item is \(\$ 200,\) and the selling price of the item is \(\$ 240\).

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-43.5| \leq 8.5,\) Punta Arenas, Chile

Solve each application of openations and composition of functions. Volume of a Sphere 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.

A circular lid is being designed for a jar. Its circumference \(C\) is designed to be 10 inches with an error tolerance of not more than 0.1 inch. (a) Write an absolute value inequality that describes all values of \(C\) that satisfy this restriction. (b) Write an absolute value inequality that gives values for the diameter \(d\) that satisfy this restriction.

An aluminum can is designed to have a height \(H\) of 5 inches with an error tolerance of not more than 0.05 inch. Write an absolute value inequality that describes all values of \(H\) that satisfy this restriction.

Solve each application of openations and composition of functions. Dimensions of a Rectangle 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 \(P\) as \(Y_{1}\) found in part (a) in the window \([0,10]\) by \([0,100]\). Locate the point for which \(x=4,\) and explain what \(x\) and \(y\) represent. (c) On the graph of \(P\), locate the point with \(x\) -value 4 . Then sketch a rectangle satisfying the conditions described carlier, 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.

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=2 x+1\) must be less than 0.1 unit from 1.

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?

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=3 x-6\) must be less than 0.3 unit from \(0 .\)

Suppose the graph of \(y=f(x)\) is symmetric with respect to the origin and is then shifted 3 units left. Is the graph still symmetric with respect to the origin?

Solve each application of operations and composition of functions. Area of an Equilateral Triangle The area \(a\) of an equilateral triangle with sides of length \(x\) is given by $$A(x)=\frac{\sqrt{3}}{4} x^{2}$$ (a) Find \(s l(2 x)\), 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 \(s l(x)\) (c) Support the result of part (b) graphically.

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=4 x-8\) must be less than 0.04 unit from 4.

Solve each application of operations and composition of functions. Emission of Pollutants \(\quad\) 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 \(8 \mathrm{AM}\).) and assume that the radius of the circle of pollution is \(r(t)=2 t\) miles. The area of a circle of radius \(r\) is represented by $$s f(r)=\pi r^{2}$$ (a) Find \((s l \circ r)(t)\) (b) Interpret \((d l \circ r)(t)\) (c) What is the area of the circular region covered by the layer at noon? (d) Support your result graphically.

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=5 x-12\) must be less than 0.1 unit from \(10 .\)

World Population and Aggregate Age The table at the top of the next column 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 \(x\). $$\begin{array}{c|c|c|c|c}\hline x & 1950 & 2000 & 2050 & 2100 \\\\\hline A(x) & 28 & 30 & 38 & 42 \\\\\hline T(x) & 80 & 180 & 360 & 430\end{array}$$ (a) Evaluate \(A(2100)\) and \(T(2100)\). Interpret your results. (b) Evaluate \(\frac{\pi 2100}{\mathrm{A}(2100)} .\) Interpret your result. (c) Let \(P(x)=\frac{\pi(x)}{A(x)}\). Interpret what \(P(x)\) calculates.

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=x-4\) must be less than 0.001 unit from 0.5.

The table below shows the acreage, in millions, of the total of com 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. $$\begin{array}{c|c|c|c|c}\hline x & 2013 & 2014 & 2015 & 2016 \\\\\hline f(x) & 175.1 & 176.4 & 174.0 & 177.8 \\\\\hline g(x) & 97.4 & 91.6 & 88.9 & 94.1\end{array}$$ (a) Make a table for a function \(h\) that is defined by the equation \(h(x)=f(x)-g(x)\) \(\Rightarrow\) (b) Interpret what \(h\) computes.

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=7-x\) must be less than 0.002 unit from 1.2.

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. $$\begin{array}{|c|c|c|c|c|c|}\hline x & 1970 & 1980 & 1990 & 2000 & 2010 \\\\\hline f(x) & 31.2 & 25.9 & 23.1 & 16.3 & 13.0 \\\\\hline g(x) & 204.0 & 185.4 & 154.2 & 114.5 & 74.3\end{array}$$ (a) Evaluate \((f+g)(2010)\) (b) Interpret \((f+g)(x)\) (c) Make a table for \((f+g)(x)\)

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

The greenhouse gas methane lets sunlight into the atmosphere, but blocks heat from escaping the earth's atmosphere. Methane is a by-product of buming 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. $$\begin{array}{|c|c|c|c|c|c|}\hline x & 1990 & 2000 & 2010 & 2020 & 2030 \\\\\hline f(x) & 27 & 28 & 29 & 30 & 31 \\\\\hline g(x) & 5 & 7.5 & 10 & 12.5 & 15\end{array}$$ (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 \(h(x)\)