Chapter 3

Exer. 21-34: Find (a) \((f \circ g)(x)\) and the domain of \(f \circ g\) and (b) \((g \circ f)(x)\) and the domain of \(g \circ f\). $$ f(x)=\frac{x+2}{x-1}, \quad g(x)=\frac{x-5}{x+4} $$

Exer. 29-34: Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. \(x\)-intercepts 8 and 0 , lowest point has \(y\)-coordinate \(-48\)

Exer. 33-36: Find the slope-intercept form of the line that satisfies the given conditions. $$ x \text {-intercept }-5, \quad y \text {-intercept }-1 $$

Exer. 23-34: Sketch the graph of the circle or semicircle. $$ x=-\sqrt{25-y^{2}} $$

Prove that the diagonals of any parallelogram bisect each other. (Hint: Label three of the vertices of the parallelogram \(O(0,0), A(a, b)\), and \(C(0, c)\).)

Exer. 33-40: Explain how the graph of the function compares to the graph of \(y=f(x)\). For example, for the equation \(y=2 f(x+3)\), the graph of \(f\) is shifted 3 units to the left and stretched vertically by a factor of 2 . $$ y=f(-x)-2 $$

Exer. 35-36: Solve the equation \((f \circ g)(x)=0\). $$ f(x)=x^{2}-2, \quad g(x)=x+3 $$

Exer. 33-36: Find the slope-intercept form of the line that satisfies the given conditions. $$ \text { Through } A(5,2) \text { and } B(-1,4) $$

Sketch the graph of a function that is increasing on \((-\infty,-3]\) and \([2, \infty)\) and is decreasing on \([-3,2]\).

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C(2,-3) \text {, radius } 5 $$

Exer. 35-36: Solve the equation \((f \circ g)(x)=0\). $$ f(x)=x^{2}-x-2, \quad g(x)=2 x-1 $$

Exer. 33-36: Find the slope-intercept form of the line that satisfies the given conditions. $$ \text { Through } A(-2,1) \text { and } B(3,7) $$

Sketch the graph of a function that is decreasing on \((-\infty,-2]\) and \([1,4]\) and is increasing on \([-2,1]\) and \([4, \infty)\).

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C(-4,1) \text {, radius } 3 $$

Several values of two functions \(f\) and \(g\) are listed in the following tables: $$ \begin{aligned} &\begin{array}{|l|lllll|} \hline \boldsymbol{x} & 5 & 6 & 7 & 8 & 9 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 8 & 7 & 6 & 5 & 4 \\ \hline \end{array}\\\ &\begin{array}{|l|lllll|} \hline \boldsymbol{x} & 5 & 6 & 7 & 8 & 9 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & 7 & 8 & 6 & 5 & 4 \\ \hline \end{array} \end{aligned} $$ If possible, find (a) \((f \circ g)(6)\) (b) \((g \circ f)(6)\) (c) \((f \circ f)(6)\) (d) \((g \circ g)(6)\) (e) \((f \circ g)(9)\)

Exer. 37-38: Ozone occurs at all levels of Earth's atmosphere. The density of ozone varies both seasonally and latitudinally. At Edmonton, Canada, the density \(D(h)\) of ozone (in \(10^{-3} \mathrm{~cm} / \mathrm{km}\) ) for altitudes \(h\) between 20 kilometers and 35 kilometers was determined experimentally. For each \(D(h)\) and season, approximate the altitude at which the density of ozone is greatest. \(D(h)=-0.058 h^{2}+2.867 h-24.239\) (autumn)

Exer. 37-38: Find a general form of an equation for the perpendicular bisector of the segment \(A B\).$$ A(3,-1), B(-2,6) $$

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=3 x-2 $$

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C\left(\frac{1}{4}, 0\right) \text {, radius } \sqrt{5} $$

Exer. 33-40: Explain how the graph of the function compares to the graph of \(y=f(x)\). For example, for the equation \(y=2 f(x+3)\), the graph of \(f\) is shifted 3 units to the left and stretched vertically by a factor of 2 . $$ y=f\left(\frac{1}{2} x\right)-3 $$

Several values of two functions \(T\) and \(S\) are listed in the following tables: $$ \begin{aligned} &\begin{array}{|l|lllll|} \hline \boldsymbol{t} & 0 & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{T}(\boldsymbol{t}) & 2 & 3 & 1 & 0 & 5 \\ \hline \end{array}\\\ &\begin{array}{|l|lllll|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{S}(\boldsymbol{x}) & 1 & 0 & 3 & 2 & 5 \\ \hline \end{array} \end{aligned} $$ If possible, find (a) \((T \circ S)(1)\) (b) \((S \circ T)(1)\) (c) \((T \circ T)(1)\) (d) \((S \circ S)(1)\) (e) \((T \circ S)(4)\)

Exer. 37-38: Ozone occurs at all levels of Earth's atmosphere. The density of ozone varies both seasonally and latitudinally. At Edmonton, Canada, the density \(D(h)\) of ozone (in \(10^{-3} \mathrm{~cm} / \mathrm{km}\) ) for altitudes \(h\) between 20 kilometers and 35 kilometers was determined experimentally. For each \(D(h)\) and season, approximate the altitude at which the density of ozone is greatest. \(D(h)=-0.078 h^{2}+3.811 h-32.433\) (spring)

Exer. 33-36: Find the slope-intercept form of the line that satisfies the given conditions. $$ A(4,2), B(-2,10) $$

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=-2 x+3 $$

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C\left(\frac{3}{4},-\frac{2}{3}\right) \text {, radius } 3 \sqrt{2} $$

Exer. 33-40: Explain how the graph of the function compares to the graph of \(y=f(x)\). For example, for the equation \(y=2 f(x+3)\), the graph of \(f\) is shifted 3 units to the left and stretched vertically by a factor of 2 . $$ y=-2 f\left(\frac{1}{3} x\right) $$

If \(D(t)=\sqrt{400+t^{2}}\) and \(R(x)=20 x\), find \((D \circ R)(x)\)

The growth rate \(y\) (in pounds per month) of an infant is related to present weight \(x\) (in pounds) by the formula \(y=c x(21-x)\), where \(c\) is a positive constant and \(0

Exer. 39-40: Find an equation for the line that bisects the given quadrants. $$ \text { II and IV } $$

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=4-x^{2} $$

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C(-4,6), \text { passing through } P(1,2) $$

If \(S(r)=4 \pi r^{2}\) and \(D(t)=2 t+5\), find \((S \circ D)(t)\).

The number of miles \(M\) that a certain automobile can travel on one gallon of gasoline at a speed of \(v \mathrm{mi} / \mathrm{hr}\) is given by $$ M=-\frac{1}{30} v^{2}+\frac{5}{2} v \quad \text { for } 0

Exer. 39-40: Find an equation for the line that bisects the given quadrants. $$ \text { I and III } $$

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=x^{2}-1 $$

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center at the origin, passing through } P(4,-7) $$

If \(f\) is an odd function and \(g\) is an even function, is \(f g\) even, odd, or neither even nor odd?

An object is projected vertically upward from the top of a building with an initial velocity of \(144 \mathrm{ft} / \mathrm{sec}\). Its distance \(s(t)\) in feet above the ground after \(t\) seconds is given by the equation $$ s(t)=-16 t^{2}+144 t+100 . $$ (a) Find its maximum distance above the ground. (b) Find the height of the building.

Exer. 41-44: Use the slope-intercept form to find the slope and \(y\)-intercept of the given line, and sketch its graph. $$ 2 x=15-3 y $$

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=\sqrt{x+4} $$

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C(-3,6) \text {, tangent to the } y \text {-axis } $$

There is one function with domain \(\mathbb{R}\) that is both even and odd. Find that function.

An object is projected vertically upward with an initial velocity of \(v_{0} \mathrm{ft} / \mathrm{sec}\), and its distance \(s(t)\) in feet above the ground after \(t\) seconds is given by the formula \(s(t)=-16 t^{2}+v_{0} t\). (a) If the object hits the ground after 12 seconds, find its initial velocity \(v_{0}\). (b) Find its maximum distance above the ground.

Exer. 41-44: Use the slope-intercept form to find the slope and \(y\)-intercept of the given line, and sketch its graph. $$ 7 x=-4 y-8 $$

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=\sqrt{4-x} $$

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C(4,-1) \text {, tangent to the } x \text {-axis } $$

Payroll functions Let the social security tax function SSTAX be defined as \(\operatorname{SSTAX}(x)=0.0765 x\), where \(x \geq 0\) is the weekly income. Let ROUND2 be the function that rounds a number to two decimal places. Find the value of \((\) ROUND2 ° SSTAX) \((525)\).

Find two positive real numbers whose sum is 40 and whose product is a maximum.

Exer. 41-44: Use the slope-intercept form to find the slope and \(y\)-intercept of the given line, and sketch its graph. $$ 4 x-3 y=9 $$

Exer. 37-46: (a) Sketch the graph of \(f\). (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing, is decreasing, or is constant. $$ f(x)=-2 $$