Chapter 3

Find the domain of the function. $$ f(x)=\frac{1}{3 x-6} $$

Express the function in the form \(f \circ g\) $$ F(x)=(x-9)^{5} $$

Find the inverse function of \(f\) $$ f(x)=\frac{1+3 x}{5-2 x} $$

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x| ;\) shift 3 units to the right and shift upward 1 unit

Find the domain of the function. $$ f(x)=\frac{x+2}{x^{2}-1} $$

Express the function in the form \(f \circ g\) $$ F(x)=\sqrt{x}+1 $$

Find the inverse function of \(f\) $$ f(x)=\frac{2 x-1}{x-3} $$

Gravity Near the Moon We can use Newton's Law of Gravity to measure the gravitational attraction between the moon and an algebra student in a space ship located a distance \(x\) above the moon's surface: $$F(x)=\frac{350}{x^{2}}$$ Here \(F\) is measured in newtons \((\mathrm{N}),\) and \(x\) is measured in millions of meters. (a) Graph the function \(F\) for values of \(x\) between 0 and \(10 .\) (b) Use the graph to describe the behavior of the gravitational attraction \(F\) as the distance \(x\) increases.

Find the domain of the function. $$ f(x)=\frac{x^{4}}{x^{2}+x-6} $$

Express the function in the form \(f \circ g\) $$ G(x)=\frac{x^{2}}{x^{2}+4} $$

Find the inverse function of \(f\) $$ f(x)=\sqrt{2+5 x} $$

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x| ;\) shift 4 units to the left and shift downward 2 units

Radii of Stars Astronomers infer the radii of stars using the Stefan Boltzmann Law: $$E(T)=\left(5.67 \times 10^{-8}\right) T^{4}$$ where \(E\) is the energy radiated per unit of surface area measured in watts \((\mathrm{W})\) and \(T\) is the absolute temperature measured in kelvins \((\mathrm{K}) .\) (a) Graph the function \(E\) for temperatures \(T\) between 100 \(\mathrm{K}\) and 300 \(\mathrm{K}\) (b) Use the graph to describe the change in energy \(E\) as the temperature \(T\) increases.

Find the domain of the function. $$ f(x)=\sqrt{x-5} $$

Express the function in the form \(f \circ g\) $$ Q(x)=\frac{1}{x+3} $$

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=x^{2} ;\) shift 2 units to the left and reflect in the \(x\) -axis

Migrating Fish A fish swims at a speed \(v\) relative to the water, against a current of 5 mifh. Using a mathematical model of energy expenditure, it can be shown that the total energy \(E\) required to swim a distance of 10 \(\mathrm{mi}\) is given by $$E(v)=2.73 v^{3} \frac{10}{v-5}$$ Biologists believe that migrating fish try to minimize the total energy required to swim a fixed distance. Find the value of \(v\) that minimizes energy required. NOTE: This result has been verified; migrating fish swim against a current at a speed 50\(\%\) greater than the speed of the current.

Find the domain of the function. $$ f(x)=\sqrt[4]{x+9} $$

Express the function in the form \(f \circ g\) $$ H(x)=\left|1-x^{3}\right| $$

Find the inverse function of \(f\) $$ f(x)=4-x^{2}, \quad x \geq 0 $$

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=x^{2} ;\) stretch vertically by a factor of \(2,\) shift downward 2 units, and shift 3 units to the right

Highway Engineering A highway engineer wants to estimate the maximum number of cars that can safely travel a particular highway at a given speed. She assumes that each car is 17 \(\mathrm{ft}\) long, travels at a speed \(s,\) and follows the car in front of it at the "safe following distance" for that speed. She finds that the number \(N\) of cars that can pass a given point per minute is modeled by the function $$N(s)=\frac{88 s}{17+17\left(\frac{s}{20}\right)^{2}}$$ At what speed can the greatest number of cars travel the highway safely?

Find the domain of the function. $$ f(t)=\sqrt[3]{t-1} $$

Express the function in the form \(f \circ g\) $$ H(x)=\sqrt{1+\sqrt{x}} $$

Find the inverse function of \(f\) $$ f(x)=\sqrt{2 x-1} $$

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x|\) shrink vertically by a factor of \(\frac{1}{2},\) shift to the left 1 unit, and shift upward 3 units

Volume of Water Between \(0^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C},\) the volume \(V\) (in cubic centimeters) of 1 \(\mathrm{kg}\) of water at a temperature \(T\) is given by the formula $$V=999.87-0.06426 T+0.0085043 T^{2}-0.0000679 T^{8}$$ Find the temperature at which the volume of 1 \(\mathrm{kg}\) of water is a minimum.

Find the domain of the function. $$ g(x)=\sqrt{7-3 x} $$

Express the function in the form \(f \circ g \circ h\) $$ F(x)=\frac{1}{x^{2}+1} $$

Find the inverse function of \(f\) $$ f(x)=4+\sqrt[3]{x} $$

Coughing When a foreign object that is lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward, causing an increase in pressure in the lungs. At the same time, the trachea contracts, causing the expelled air to move faster and increasing the pressure on the foreign object. According to a mathematical model of coughing, the velocity \(v\) of the air stream through an average-sized person's trachea is related to the radius \(r\) of the trachea (in centimeters) by the function $$v(r)=3.2(1-r) r^{2} \quad \frac{1}{2} \leq r \leq 1$$ Determine the value of \(r\) for which \(v\) is a maximum.

Find the domain of the function. $$ h(x)=\sqrt{2 x-5} $$

Express the function in the form \(f \circ g \circ h\) $$ F(x)=\sqrt[3]{\sqrt{x}-1} $$

Find the inverse function of \(f\) $$ f(x)=\left(2-x^{3}\right)^{5} $$

Functions That Are Always Increasing or Decreasing Sketch rough graphs of functions that are defined for all real numbers and that exhibit the indicated behavior (or explain why the behavior is impossible). (a) \(f\) is always increasing, and \(f(x)>0\) for all \(x\) (b) \(f\) is always decreasing, and \(f(x)>0\) for all \(x\) (d) \(f\) is always decreasing, and \(f(x)<0\) for all \(x\)

Find the domain of the function. $$ G(x)=\sqrt{x^{2}-9} $$

Express the function in the form \(f \circ g \circ h\) $$ Q(x)=(4+\sqrt[3]{x})^{9} $$

Find the inverse function of \(f\) $$ f(x)=1+\sqrt{1+x} $$

Find the domain of the function. $$g(x)=\frac{\sqrt{2+x}}{3-x}$$

Express the function in the form \(f \circ g \circ h\) $$ G(x)=\frac{2}{(3+\sqrt{x})^{2}} $$

Find the inverse function of \(f\) $$ f(x)=\sqrt{9-x^{2}}, \quad 0 \leq x \leq 3 $$

Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose $$g(x)=\sqrt{f(x)}$$ where \(f(x) \geq 0\) for all \(x\) Explain why the local minima and maxima of \(f\) and \(g\) occur at the same values of \(x .\) (b) Let \(g(x)\) be the distance between the point \((3,0)\) and the point \(\left(x, X^{2}\right)\) on the graph of the parabola \(y=x^{2} .\) Express \(g\) as a function of \(x\) (c) Find the minimum value of the function \(g\) that you found in part (b). Use the principle described in part (a) to simplify your work.

Find the domain of the function. $$ g(x)=\frac{\sqrt{x}}{2 x^{2}+x-1} $$

Revenue, cost, and Profit A print shop makes bumper stickers for election campaigns. If \(x\) stickers are ordered (where \(x<10,000\) ), then the price per bumper sticker is \(0.15-0.000002 x\) dollars, and the total cost of producing the order is \(0.095 x-0.0000005 x^{2}\) dollars. Use the fact that revenue \(=\) price per tiem \(\times\) number of items sold to express \(R(x),\) the revenue from an order of \(x\) stickers, as a product of two functions of \(x\)

Determine whether the equation defines y as a function of x. (See Example 9.) \(x=y^{2}\)

Find the inverse function of \(f\) $$ f(x)=x^{4}, \quad x \geq 0 $$

Find the domain of the function. $$ g(x)=\sqrt[4]{x^{2}-6 x} $$

Revenue, cost, and Profit A print shop makes bumper stickers for election campaigns. If \(x\) stickers are ordered (where \(x<10,000\) ), then the price per bumper sticker is \(0.15-0.000002 x\) dollars, and the total cost of producing the order is \(0.095 x-0.0000005 x^{2}\) dollars. Use the fact that $$ \text { profit }=\text { revenue }-\text { cost } $$ to express \(P(x)\) , the profit on an order of \(x\) stickers, as a difference of two functions of \(x .\)

Determine whether the equation defines y as a function of x. (See Example 9.) \(x^{2}+(y-1)^{2}=4\)

Find the inverse function of \(f\) $$ f(x)=1-x^{3} $$