Chapter 3

Solve. Write answers in standard form. $$ 3 x=5 x^{2}+1 $$

Find the domain of the function. Write your answer in set-builder notation. $$ f(x)=\frac{1}{x^{2}-5} $$

The stopping distance \(d\) in feet for a car traveling at \(x\) miles per hour is given by \(d(x)=\frac{1}{12} x^{2}+\frac{11}{9} x\). Determine the driving speeds that correspond to stopping distances between 300 and 500 feet, inclusive. Round speeds to the nearest mile per hour.

Use transformations to sketch a graph of \(f\). \(f(x)=\sqrt{-x}-1\)

Solve. Write answers in standard form. $$ 4 x^{2}=x-1 $$

Find the domain of the function. Write your answer in set-builder notation. $$ f(x)=\frac{4 x}{7-x^{2}} $$

The volume of a cylinder is given by \(V=\pi r^{2} h,\) where \(r\) is the radius and \(h\) is the height. If the height of a cylindrical can is 6 inches and the volume must be between \(24 \pi\) and \(54 \pi\) cubic inches, inclusive, find the possible values for the radius of the can

Solve. Write answers in standard form. $$ x(x-4)=-5 $$

Find the domain of the function. Write your answer in set-builder notation. $$ g(t)=\frac{5-t}{t^{2}-t-2} $$

A rectangle is 4 feet longer than it is wide. If the area of the rectangle must be less than or equal to 672 square feet, find the possible values for the width \(x\).

Use transformations to sketch a graph of \(f\). \(f(x)=2+\sqrt{-(x-3)}\)

Solve. Write answers in standard form. $$ 2 x^{2}+x+1=0 $$

Find the domain of the function. Write your answer in set-builder notation. $$ g(t)=\frac{t+1}{2 t^{2}-11 t-21} $$

Suppose that a person's heart rate, \(x\) minutes after vigorous exercise has stopped, can be modeled by \(f(x)=\frac{4}{5}(x-10)^{2}+80\). The output is in beats per minute, where the domain of \(f\) is \(0 \leq x \leq 10\). (a) Evaluate \(f(0)\) and \(f(2)\). Interpret the result. (b) Estimate the times when the person's heart rate was between 100 and 120 beats per minute, inclusive.

Use transformations to sketch a graph of \(f\). \(f(x)=(x-1)^{3}\)

Solve. Write answers in standard form. $$ x^{2}=3 x-5 $$

Solve the equation for \(y .\) Determine if y is a function of \(x\). $$ 4 x^{2}+3 y=\frac{y+1}{3} $$

Exposure When a person breathes carbon monoxide (CO), it enters the bloodstream to form carboxyhemoglobin (COHb), which reduces the transport of oxygen to tissues. The formula given by \(T(x)=0.0078 x^{2}-1.53 x+76\) approximates the number of hours \(T\) that it takes for a person's bloodstream to reach the \(5 \%\) COHb level, where \(x\) is the concentration of \(\mathrm{CO}\) in the air in parts per million (ppm) and \(50 \leq x \leq 100\). (Smokers routinely have a \(5 \%\) concentration.) Estimate the CO concentration \(x\) necessary for a person to reach the \(5 \%\) COHb level in \(4-5\) hours.

Use transformations to sketch a graph of \(f\). \(f(x)=(x+2)^{3}\)

Solve. Write answers in standard form. $$ 3 x-x^{2}=5 $$

Solve the equation for \(y .\) Determine if y is a function of \(x\). $$ \frac{x^{2}+y}{2}=y-2 $$

Use transformations to sketch a graph of \(f\). \(f(x)=-x^{3}\)

Solve. Write answers in standard form. $$ x^{2}+2 x+4=0 $$

Solve the equation for \(y .\) Determine if y is a function of \(x\). $$ 3 y=\frac{2 x-y}{3} $$

As the altitude increases, air becomes thinner, or less dense. An approximation of the density of air at an altitude of \(x\) meters above sea level is given by \(d(x)=\left(3.32 \times 10^{-9}\right) x^{2}-\left(1.14 \times 10^{-4}\right) x+1.22\) The output is the density of air in kilograms per cubic meter. The domain of \(d\) is \(0 \leq x \leq 10,000\). (a) Denver is sometimes referred to as the mile-high city. Compare the density of air at sea level and in Denver. (Hint: 1 ft \(\approx 0.305\) m.) (b) Determine the altitudes where the density is greater than 1 kilogram per cubic meter.

Use transformations to sketch a graph of \(f\). \(f(x)=(-x)^{3}+1\)

Solve. Write answers in standard form. $$ x(x-4)=-8 $$

Solve the equation for \(y .\) Determine if y is a function of \(x\). $$ \frac{5-y}{3}=\frac{x+3 y}{4} $$

For the given representation of a function \(f,\) graph the reflection across the \(x\)-axis and graph the reflection across the \(y\)-axis. \(f(x)=x^{2}-2 x-3\)

Solve. Write answers in standard form. $$ 3 x^{2}-4 x=x^{2}-3 $$

Solve the equation for \(y .\) Determine if y is a function of \(x\). $$ x^{2}+(y-3)^{2}=9 $$

For the given representation of a function \(f,\) graph the reflection across the \(x\)-axis and graph the reflection across the \(y\)-axis. \(f(x)=4-7 x-2 x^{2}\)

Solve. Write answers in standard form. $$ 2 x^{2}+3=1-x $$

Solve the equation for \(y .\) Determine if y is a function of \(x\). $$ (x+2)^{2}+(y+1)^{2}=1 $$

Explain how a table of values can be used to help solve a quadratic inequality, provided that the boundary numbers are listed in the table.

For the given representation of a function \(f,\) graph the reflection across the \(x\)-axis and graph the reflection across the \(y\)-axis. \(f(x)=|x+1|-1\)

Solve. Write answers in standard form. $$ 2 x(x-2)=x-4 $$

Solve the equation for \(y .\) Determine if y is a function of \(x\). $$ 3 x^{2}+4 y^{2}=12 $$

Explain how to determine the solution set for the inequality \(a x^{2}+b x+c<0,\) where \(a>0 .\) How would the solution set change if \(a<0 ?\)

For the given representation of a function \(f,\) graph the reflection across the \(x\)-axis and graph the reflection across the \(y\)-axis. \(f(x)=\frac{1}{2}|x-2|+2\)

Solve. Write answers in standard form. $$ 3 x^{2}+x-x(5-x)-2 $$

Solve the equation for \(y .\) Determine if y is a function of \(x\). $$ x-25 y^{2}=50 $$

For the given representation of a function \(f,\) graph the reflection across the \(x\)-axis and graph the reflection across the \(y\)-axis. Line graph determined by the table \(\begin{array}{rrrrr}x & -3 & -1 & 1 & 2 \\ f(x) & 2 & 3 & -1 & -2\end{array}\)

Solve. Write answers in standard form. $$ 3 x(3-x)-8=x(x-2) $$

Solve for the specified variable. $$ V=\frac{1}{3} \pi r^{2} h \text { for } r $$

For the given representation of a function \(f,\) graph the reflection across the \(x\)-axis and graph the reflection across the \(y\)-axis. Line graph determined by the table \(\begin{array}{rrrrr}x & -4 & -2 & 0 & 1 \\ f(x) & -1 & -4 & -2 & -2\end{array}\)

Solve. Write answers in standard form. $$ -x(7-2 x)=-6-(3-x) $$

Solve for the specified variable. $$ V=\frac{1}{2} g t^{2}+h \text { for } t $$

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x)+7\) $$\begin{array}{rrrrrrr}x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline f(x) & 5 & 1 & 6 & 2 & 7 & 9\end{array}$$

Solve for the specified variable. $$ \boldsymbol{K}=\frac{1}{2} m \boldsymbol{v}^{2} \text { for } \boldsymbol{v} $$