Week 4 Pre-calculus Reading Assignment

- Ch 3.4 pp 188-190 through the try-it-now 1 and then 191 through Example 3
- Ch 3.5 pp 206-208 through the try-it-now 1 and p 210 Example 6.
- Ch 1.1, Ch 1.2 pp21-22, 24-26, 1.6 pp90-93 (skip Example 2 p91) as needed for background for the Ch 3 reading.

Week 4 Reading Guide

This week’s short precalculus reading concentrates narrowly on terminology and background concepts that set up our work in Math Lab 4 and in coming weeks. Such need-directed reading is common in science and our case requires side-trips into Ch 1 for background definitions and concepts. This reading guide is intended to suggest a sequence of reading to cover background excursions.

- Read Ch 3.4 pp 188-189. Study the definitions of vertical and horizontal asymptotes at the bottom of p189 and connect these concepts with the graphs at the top of p189 by closely reading the middle of p189 and then doing the try-it-now at the top of p190. Practice reading the notation for behavior around asymptotes. Asymptotes are associated with what the text calls long-run (as x very large) and short-run behavior (as x gets very close to an undefined x value). The graphs and the symbolic notation should tell the story, but If you want background reading on the terms long-run and short-run, you can review the definitions of terms in Ch 3.1.
- Do the Try-it-now at the top of p190 for practice with the concepts and the terms.
- Read the top half of p191. Connect the example with the definition at the top of the page.
- Read through the Example 1 p206-207. The example does some algebra with which you are familiar – solve the function f(x) = ½ x
^{2}for x (read the second-to-last sentence at the bottom of p206). Remember that solving is reverse evaluation. The Example 1 shows how you can build a new function that directly calculates the reverse evaluation. The only trick is that you have to throw away all but a single result. Notice how the example does that – it throws away the negative root. When you construct a new function as a reverse evaluation of another function you are constructing an inverse function. - The discussion in Example 1 p206-207 relies on the general concept of the inverse of a function and the concept of a one-to-one function. The concepts of a one-to-one function is introduced in Ch 1.1 and the concept of an inverse of a function is discussed in Ch 1.6. Now is a good time to read those sections (see the supplementary Reading Guide for those sections below) and work to connect the general concepts to the discussion in Example 1 p206-207. You may also need to read some parts of Ch 1.2 on domain and range. See the supplementary reading guide below covering Ch 1.1, 1.2, and 1.6. Also you can review your work on Math Lab 2 where domain and range were introduced and where you graphed the toolkit functions (in the last part of the lab). Be prepared with questions for Monday’s class.
- Do the algebra in Example 2 p207-208 to find the inverse function of the given function f. Notice that the inverse function for f has the name f
^{-1}. The minus 1 is part of the name of the function, not an operation. Inverse functions have the fun property that they completely undo the action of a function, that is, if you apply f^{-1}to f(x) you get back x. It works the other way too! If you apply f to f^{-1}(y) you get back y. For example f ( f^{-1}(3)) = 3 and f^{-1}( f(3) ) = 3 - To get a another picture of the undo property of inverse functions, go back to Example 1 p 206 where the inverse of the function f(x) = ½ x
^{2}is f^{-1}(y) = sqrt (2 * y); note: sqrt (blah) means take the square root of blah. Now calculate f (f^{-1}(y)) = f (sqrt (2*y)) = ½ (sqrt (2*y))^{-2}. . Now calculate f^{-1}( f(x) ). Amazing, eh? - If the last three steps of this reading guide are confusing, don’t worry – we’ll do them and talk about them in class as needed. The concepts are simple, but the notation can be confusing if you’re not used to it.
- Study Example 6 p210. Notice that the problem really just builds the inverse function so reverse evaluation (solving) can be easily done by a calculation!
- Do the Try-it-now 3 at the bottom of p210.

Reading Guide for Ch 1.1, 1.2, 1.6 (supplementary)

- Most of Ch 1.1 may now be familiar to you from our previous work. The key ideas are:
- Functions are rules that allow you to calculate outputs given inputs.
- There are three common ways to represent functions: tables, graphs, and formulas.
- To be called a function, a rule MUST have a single output for a given input.
- Evaluation is calculating the value of a function
- Solving a function is reverse-evaluating a function.
- A one-to-one function is a function that gives a unique answer when reverse evaluating. Note that quadratics are not one-to-one because they can give more than one answer upon reverse evaluation.

- At the end of Ch 1.1 look at the formulas and graphs for the toolkit functions. You did this in Math lab 2; now is a GREAT time to finish the last step of Math lab 2 if you have not already.
- Ch 1.2 is all about establishing the sensible inputs and outputs for a function given the context of the function. You have already seen this in various problems. You can focus on the key ideas we’ll need by reading the following:
- Read pp21-22 for definitions of Domain and Range. You encountered domain and range in Math lab 2 when you played with domain and range in Desmos.
- Read about domain and range for graphs on pp 24-26 through the Try-it-now 3.

- Ch 1.6 on inverse functions is fun! I suggest reading the whole section. But for our specific purposes we’ll mainly need the definitions and properties of inverse functions on pp 90-93. Skip Example 2 p91. Practice your algebra by solving for the Celsius to Fahrenheit function given the Fahrenheit to Celsius function at the beginning of Ch 1.6. You’ll really help Betty out!