Lecture 17, Composition of Functions
Wednesday, March 3, 1999

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[ List of Lectures | Math 1100 Index ]

Problem of the day

This problem is due on Friday, March 5:
POW 8

Assignments during second test period.

These are the problems you should work before March 5.

Section 2.2 page 205 problems 8n+1, for n=0,...,11;

Section 2.3 page 218 problems 1-6 and 6n+1, for n=1,...,8;

Section 2.4 page 232 problems 1-6 and 4n+1, for n=2,…,22;

These are the problems you should work before March 19.

Section 2.5 page 244 problems 1-6 and 6n+1, for n=1,...,14;

Review page 249 problems 4n+1, n=0,…,14;

Section 3.1 page 263 problems 1-6 and 4n+1, for n=0,…,19;

These are the problems you should work before March 26.

Section 3.2 page 277 problems 4n+1, n=0,...,12;

Section 3.3 page 289 problems 4n+1, n=0,...,7;

Section 3.4 page 298 problems 4n+1, n=2,...,7

 Deadlines for submitting the ICA tutorial software post-tests:
The policy for late submission is that the student loses 5 points per week for each posttest which is submitted late. The loss begins the day following the deadline.

Brief review of the lecture

The single most important operation on functions is composition of functions. There are many ways to form new functions out of old ones, like adding them, multiplying them, or subtracting them. These are operations we can perform on all real numbers, so we can easily understand these operations. But composition of functions is an operation that we can only perform on functions. You'll see in the calculus course a formula called the Chain Rule. Its purpose is to enable you to find the derivative of a function which is build as a composition. In order to understand the use of this important rule, you must understand composition of functions very well. For that reason, we'll spend more than one full lecture discussing this operation. For a list of skills required for the test, click here

Leftovers

Below is the latex code which produced the practice test. It is not easy to read, so picking up a copy at Dr. Reiter's office is recommended. \item The first three questions apply to the table given below: Suppose the functions $f$ and $g$ are given completely by the table of values shown. \newline \centerline { \begin{tabular}{c|c|c} $x$& $f(x)$&$g(x)$ \\ \hline 0 & 2 & 5 \\ 1 & 7 &7 \\ 2 & 6 &4 \\ 3 & 1 &2 \\ 4 & 3 &4 \\ 5 & 5 &3 \\ 6 & 6 &4 \\ 7 & 4 &0 \end{tabular}} \vfill \begin{enumerate} \item What is the value of $f(g(f(3)$? \ansmt{1}{3}{4}{6}{7}\vfill \item Given that $(f(x-3))=7$, what is $g(x)$? \ansmt{1}{3}{4}{6}{7}\vfill \item What is $f(g(1+2)+g(6-1))$? \ansmt{1}{3}{4}{5}{7}\vfill \end{enumerate} \item The equation $x^2-2x+y^2+6y=6$ describes a circle with center at $(h,k)$ and radius $r$. Find $h+k+r$. \ansmt{2}{8}{9}{14}{20}\vfill \item The sum of the roots of $|x|=x^2+x-3$ is \ansmt{0}{3-\sqrt 3}{\sqrt 3}{3}{\sqrt 3-3} \item How many integers $x$ satisfy $(x-{\frac 92})(x-{\frac 52}) (x+{\frac 12})(x+{\frac 92})<0$? \ansmt{1}{2}{4}{6}{7}\vfill \item The four numbers $\frac 14, x, y, \mbox{ and }\frac 23$ are in order from smallest to largest and are equally spaced. What is $x$? \ansmt{\frac{13}{23}}{\frac{7}{18}}{\frac{29}{36}}{\frac{5}{12}} {\frac13}\vfill \item What is the value of $2-(1-(2-(1-(2-(1-(2))))))$? \ansmt{-3}{-6}{3}{4}{5} \item Two functions $f$ and $g$ are defined by $f(x)=2x+1$ and $g(x)=x^2-2$. At what points do their graphs intersect? \ansmtXXX{(3,1)\mbox{ and }(-3,-7)}{(1,7)\mbox{ and }(3,1)} {(-1,-1)\mbox{ and }(3,7)}{(7,1)\mbox{ and }(-3,-1)} {\mbox{They do not intersect.}} \item The average (arithmetic mean) of the roots of $2x^2+14x+17=0 \mbox{ is }$ \ansmt{-7}{-7/2}{\sqrt 15/9}{7/2}{17/4} \item If $a>0$ and $b<0$, which of the following must be true? \ansmt{a>-b}{-a>b}{a-b>0}{-a>-b}{ab>0} \item Which of the following equations has exactly the same roots as $|x+4|=7$? \ansmtXXX{x^2+14x+33=0}{x^2-14x+33=0}{x^2-8x-33=0}{x^2+8x-33=0}{x^2+33x-14=0} \item Which of the following statements is true for all values (positive, negative, or zero) of the number c? \ansmtXXX{8c>4c}{4c>8c}{8c^2>4c^2}{8+c>4+c}{8+4c>4-4c} \item Consider the line segment shown: %\newline %\centerline { \beginpicture \setcoordinatesystem units < 0.30000in, 0.35000in> \setlinear \putrule from 0.00000 0.00000 to 6.00000 0.00000 \put {$u$}<0pt,8pt> at 1.00000 0.00000 \put {$u$}<0pt,8pt> at 3.00000 0.00000 \put {$u$}<0pt,8pt> at 5.00000 0.00000 \put {$\scriptstyle\bullet$} at 0.00000 0.00000 \put {$P$}<0pt,8pt> at 0.00000 0.00000 \put {$\scriptstyle\bullet$} at 6.00000 0.00000 \put {$Q$}<0pt,8pt> at 6.00000 0.00000 \put {$\scriptstyle\bullet$} at 2.00000 0.00000 \put {$R$}<0pt,8pt> at 2.00000 0.00000 \put {$\scriptstyle\bullet$} at 4.00000 0.00000 \put {$S$}<0pt,8pt> at 4.00000 0.00000 \endpicture} \newline Which of the following is true? \ansmtXXX {Q=\frac{1}{2}P+\frac 12 R}{R=\frac{2}{3}Q+\frac{1}{3}S} {Q=\frac{2}{3}S+\frac{1}{3}P} {Q=\frac{1}{3}S+\frac{2}{3}P} {R=\frac{1}{2}(P+S)} \item Let $A$ be the point $(3,2)$. Let $B$ be the result of reflecting $A$ through the $y$-axis. Let $C$ be the reflection of $B$ through the line $y=x$.Which of the following is closest to the distance between $A$ and $C$. \ansmt{2.44}{4.80}{5.10}{5.21}{5.33}\vfill \item If $S$ is the sum of the roots of $2x^2-3x+5=0$ and $P$ is the product of the roots, the value of $S-P$ is \ansmt{-1}{0}{1}{2}{4} \item Anna starts at a point $A$ and walks north three miles, then walks east 2 miles, then south 1 mile, then east 4 miles, and finally she walks north again 2 miles, arriving at point $B$. Which of the following is closest to the distance between point $A$ and point $B$? \ansmt{5.8}{6.3}{7.2}{10}{12} \newpage % \item What is the shortest distance between the origin and the point on $4x^2+4y^2+20x-16y-23=0$ which is closest to the origin? \ansmt{\sqrt{\frac{41}{4}}} {\sqrt{2-\frac{41}{4}}} {\sqrt{\frac{41}{4}}-1} {\sqrt{\frac{41}{2}}} {4-\sqrt{\frac{41}{4}}} \item The inequality $(x+1)(x-3)(x-5)>0$ is satisfied for all $x$ in which set? \ansmtXXX{-1 2$} \end{array} \right . $. \newline \begin{itemize} \item Evaluate $f(-2)$ and $f(3)$. \item Sketch the graph of $f$. \end{itemize} \item Consider the fragment of programming code below: %\par \vspace{.5in} \vbox { \parbox{4in} \newline {\tt \phantom{aaaaaaaaaa} If $X < 3 \mbox{ then } X:= 2*X + 1 \mbox{ else } X:=X-4$; \newline \phantom{aaaaaaaaaa} If $X< 5 \mbox{ then } X:=X-5 \mbox{ else } X:= X*X$} \vspace{.2in} \begin{enumerate} \item Suppose $X$ is a real variable. Find two functions $f$ and $g$ which describe the effect of each {\tt If then else} statement on variable $X$. \item Now use the functions from part (a) to find the effect on $X$ of the {\em entire} fragment of code. \end{enumerate} \item The driver of a large truck traveled at an average speed of 55 miles per hour on a 200 mile trip to pick up a load of freight. On the return trip (with the truck fully loaded), the average speed was 40 miles per hour. Find the average speed for the round trip. Hint: How much total time did the entire 400 mile trip take? \item Find the vertical and horizontal asymptotes of $$g(x)= \frac{5x-3}{2x+7}$$ and sketch the graph. \item Let $f$ be the function defined by $f(x)=\left\{ \begin{array}{ll} x^2+1 & \mbox{if $x<0$} \\ \sqrt{4-x} & \mbox{if $x\geq 0$} \end{array} \right . $. \newline Describe the domain of $f$. \item Let $f$ and $g$ be functions defined by $$f(x)=\left\{ \begin{array}{ll} x^2+1 & \mbox{if $x<0$} \\ 4-x & \mbox{if $x\geq 0$} \end{array} \right . \mbox{ and } g(x)=\left\{ \begin{array}{ll} 2x & \mbox{if $x<5$} \\ 3x & \mbox{if $x\geq 5$} \end{array} \right . $$% \newline Notice that $g(f(1))=g(3)=6$. Find a representation of the function $g(f(x))$. This will require three clauses(pieces). Simplify the representation by so that each clause is determined by one inequality. \item Solve each of the following equations and inequalities for x: \begin{enumerate} \item $\frac 12\leq 2x-2\leq \frac 32$ \item $\left| 2x-4\right| \geq 1$ \item $\frac x{x^2-4}+\frac 1{x+2}=3$

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