Chapter 1

change each repeating decimal to a ratio of two integers. $$ 2.56565656 \ldots $$

Find the value of \(c\) for which the line \(3 x+c y=5\) (a) passes through the point \((3,1)\); (b) is parallel to the \(y\) -axis; (c) is parallel to the line \(2 x+y=-1\); (d) has equal \(x\) - and \(y\) -intercepts; (e) is perpendicular to the line \(y-2=3(x+3)\).

Find a formula for \(f^{-1}(x)\) and then verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\) $$ f(x)=x^{5 / 2}, x \geq 0 $$

Let \(\ell_{1}\) and \(\ell_{2}\) be two nonvertical intersecting lines with slopes \(m_{1}\) and \(m_{2}\), respectively. If \(\theta\), the angle from \(\ell_{1}\) to \(\ell_{2}\), is not a right angle, then $$ \tan \theta=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} $$

Let \(A(c)\) denote the area of the region bounded from above by the line \(y=x+1\), from the left by the \(y\) -axis, from below by the \(x\) -axis, and from the right by the line \(x=c .\) Such a function is called an accumulation function. (See Figure 13.) Find (a) \(A(1)\) (b) \(A(2)\) (c) \(A(0)\) (d) \(A(c)\) (e) Sketch the graph of \(A(c)\). (f) What are the domain and range of \(A\) ?

.\( Find the distance between the points on the circle \)x^{2}+y^{2}=13\( with the \)x\( -coordinates \)-2\( and \)2 .$ How many such distances are there?

Find the solution sets of the given inequalities. $$ \left|\frac{x}{4}+1\right|<1 $$

change each repeating decimal to a ratio of two integers. $$ \text { 3.929292... } $$

41\. Write the equation for the line through \((-2,-1)\) that is perpendicular to the line \(y+3=-\frac{2}{3}(x-5)\)

Find a formula for \(f^{-1}(x)\) and then verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\) $$ f(x)=\frac{x-1}{x+1} $$

Let \(B(c)\) denote the area of the region bounded from above by the graph of the curve \(y=x(1-x)\), from below by the \(x\) -axis, and from the right by the line \(x=c\). The domain of \(B\) is the interval \([0,1]\). (See Figure 14.) Given that \(B(1)=\frac{1}{6}\), (a) Find \(B(0)\) (b) Find \(B\left(\frac{1}{2}\right)\) (c) As best you can, sketch a graph of \(B(c)\).

. Find the distance between the points on the circle \(x^{2}+2 x+y^{2}-2 y=20\) with the \(x\) -coordinates \(-2\) and 2. How many such distances are there?

Find the solution sets of the given inequalities. $$ |5 x-6|>1 $$

change each repeating decimal to a ratio of two integers. $$ 0.199999 \ldots $$

Find the value of \(k\) such that the line \(k x-3 y=10\) (a) is parallel to the line \(y=2 x+4\); (b) is perpendicular to the line \(y=2 x+4\); (c) is perpendicular to the line \(2 x+3 y=6\).

Find a formula for \(f^{-1}(x)\) and then verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\) $$ f(x)=\left(\frac{x-1}{x+1}\right)^{3} $$

Which of the following functions satisfies \(f(x+y)=f(x)+f(y)\) for all real numbers \(x\) and \(y\) ? (a) \(f(t)=2 t\) (b) \(f(t)=t^{2}\) (c) \(f(t)=2 t+1\) (d) \(f(t)=-3 t\)

Find the solution sets of the given inequalities. $$ |2 x-7|>3 $$

change each repeating decimal to a ratio of two integers. $$ 0.399999 \ldots $$

\text { Does }(3,9) \text { lie above or below the line } y=3 x-1 ?

Find a formula for \(f^{-1}(x)\) and then verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\) $$ f(x)=\frac{x^{3}+2}{x^{3}+1} $$

. Find the area of the sector of a circle of radius 5 centime ters and central angle 2 radians (see Problem 42).

Let \(f(x+y)=f(x)+f(y)\) for all \(x\) and \(y\). Prove that there is a number \(m\) such that \(f(t)=m t\) for all rational numbers t. Hint: First decide what \(m\) has to be. Then proceed in steps, starting with \(f(0)=0, f(p)=m p\) for a natural number \(p\), \(f(1 / p)=m / p\), and so on.

Find the solution sets of the given inequalities. $$ \left|\frac{1}{x}-3\right|>6 $$

Since \(0.199999 \ldots=0.200000 \ldots\) and \(0.399999 \ldots=\) \(0.400000 \ldots\) (see Problems 41 and 42), we see that certain rational numbers have two different decimal expansions. Which rational numbers have this property?

Find a formula for \(f^{-1}(x)\) and then verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\) $$ f(x)=\left(\frac{x^{3}+2}{x^{3}+1}\right)^{5} $$

A regular polygon of \(n\) sides is inscribed in a circle of radius \(r\). Find formulas for the perimeter, \(P\), and area, \(A\), of the polygon in terms of \(n\) and \(r\).

A baseball diamond is a square with sides of 90 feet. A player, after hitting a home run, loped around the diamond at 10 feet per second. Let \(s\) represent the player's distance from home plate after \(t\) seconds. (a) Express \(s\) as a function of \(t\) by means of a four-part formula. (b) Express \(s\) as a function of \(t\) by means of a three-part formula.

Find the solution sets of the given inequalities. $$ \left|2+\frac{5}{x}\right|>1 $$

. Show that any rational number \(p / q\), for which the prime factorization of \(q\) consists entirely of \(2 \mathrm{~s}\) and \(5 \mathrm{~s}\), has a terminating decimal expansion.

In Problems 45-48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(2 x+3 y=4\) \(-3 x+y=5\)

To use technology effectively, you need to discover its capabilities, its strengths, and its weaknesses. We urge you to practice graphing functions of various types using your own computer package or calculator. Let \(f(x)=\left(x^{3}+3 x-5\right) /\left(x^{2}+4\right)\). (a) Evaluate \(f(1.38)\) and \(f(4.12)\). (b) Construct a table of values for this function corresponding to \(x=-4,-3, \ldots, 3,4\).

Solve the given quadratic inequality using the Quadratic Formula. $$ x^{2}-3 x-4 \geq 0 $$

Find a positive rational number and a positive irrational number both smaller than \(0.00001\).

In Problems 45-48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(4 x-5 y=8\) \(2 x+y=-10\)

A ball is thrown vertically upward with velocity \(v_{0}\). Find the maximum height \(H\) of the ball as a function of \(v_{0}\). Then find the velocity \(v_{0}\) required to achieve a height of \(H .\) Hint: The height of the ball after \(t\) seconds is \(h=-16 t^{2}+v_{0} t .\) The vertex of the parabola \(y=-a x^{2}+b x\) is at \(\left(b /(2 a), b^{2} /(4 a)\right)\).

From a product identity, we obtain $$ \cos \frac{x}{2} \cos \frac{x}{4}=\frac{1}{2}\left[\cos \left(\frac{3}{4} x\right)+\cos \left(\frac{1}{4} x\right)\right] $$

Solve the given quadratic inequality using the Quadratic Formula. $$ x^{2}-4 x+4 \leq 0 $$

What is the smallest positive integer? The smallest positive rational number? The smallest positive irrational number?

In Problems 45-48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(3 x-4 y=5\) \(2 x+3 y=9\)

The normal high temperature for Las Vegas, Nevada, is \(55^{\circ} \mathrm{F}\) for January 15 and \(105^{\circ}\) for July 15 . Assuming that these are the extreme high and low temperatures for the year, use this information to approximate the average high temperature for November 15 .

To use technology effectively, you need to discover its capabilities, its strengths, and its weaknesses. We urge you to practice graphing functions of various types using your own computer package or calculator. Draw the graph of \(f(x)=x^{3}-5 x^{2}+x+8\) on the domain \([-2,5]\). (a) Determine the range of \(f\). (b) Where on this domain is \(f(x) \geq 0\) ?

Solve the given quadratic inequality using the Quadratic Formula. $$ 14 x^{2}+11 x-15 \leq 0 $$

$$ \begin{array}{l} \text { Find a rational number between } 3.14159 \text { and } \pi . \text { Note that }\\\ \pi=3.141592 \ldots \end{array} $$

In Problems 45-48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(5 x-2 y=5\) \(2 x+3 y=6\)

. Tides are often measured by arbitrary height markings at some location. Suppose that a high tide occurs at noon when the water level is at 12 feet. Six hours later, a low tide with a water level of 5 feet occurs, and by midnight another high tide with a water level of 12 feet occurs. Assuming that the water level is periodic, use this information to find a formula that gives the water level as a function of time. Then use this function to approximate the water level at \(5: 30 \mathrm{P} . \mathrm{M}\).

Solve the given quadratic inequality using the Quadratic Formula. $$ 14 x^{2}+11 x-15 \leq 0 $$

. Is there a number between \(0.9999 \ldots\) (repeating \(9 \mathrm{~s}\) ) and 1? How do you resolve this with the statement that between any two different real numbers there is another real number?

The points \((2,3),(6,3),(6,-1)\), and \((2,-1)\) are corners of a square. Find the equations of the inscribed and circumscribed circles.

Classify each of the following as a PF (polynomial function), RF (rational function but not a polynomial function), or neither. (a) \(f(x)=3 x^{1 / 2}+1\) (b) \(f(x)=3\) (c) \(f(x)=3 x^{2}+2 x^{-1}\) (d) \(f(x)=\pi x^{3}-3 \pi\) (e) \(f(x)=\frac{1}{x+1}\) (f) \(f(x)=\frac{x+1}{\sqrt{x+3}}\)