MATH 1113 Review Sheet for Test #3 (Chapter 5)

Section 5.1 Composition

·  Defining a composition of functions in words: Given functions f and g, the composition of f with g, denoted f o g is the function defined by (f o g)(x) = f(g(x)).

·  Intuitive idea of "chaining" functions together

·  Finding the composition of functions using formula descriptions

·  Determining the domain of the composition

·  Evaluating the composition of functions at a point

·  Decomposing a composition into component functions

Section 5.2 Inverse Functions

·  Defining Reverses and Inverses of a function

·  Inverse Function of a function; connection between domains and ranges of these functions

·  Defining the terms one to one and one to one function

o  Intuitively, one to one means no partner sharing

o  Determining when a graph that represents a function is one to one; the horizontal line test

·  Relationship between a one to one function and its inverse function in terms of composition

·  Relationship between the graphs of a one to one function and its inverse function

·  Reading the domain and range of a function from the graph

·  Finding the inverse function of a one to one function using formula descriptions

·  Finding the range of a one to one function by finding the domain of its inverse function

Section 5.3 Exponential Functions

·  Laws of exponents; add this law ax-y = ax/ay for all positive real numbers a, and for all real numbers x and y.

·  Definition of an exponential function; restrictions on the bases we consider; all have vertical intercept (0,1)

·  Graphs of exponential functions: two cases (0 < a < 1 (exponential decay) ; a >1 (exponential growth))

·  Transformations (scaling, reflecting, shifting) of exponential functions

·  Definition of the irrational number e in terms of a limit, e is considered the natural base

·  Characterization of exponential functions:

o  If E(x)=ax is an exponential function, then E(d)/E(c) = ad-c for all real numbers c and d.

·  Exponential functions are one to one

·  Simple exponential equations – strategy: write each side as an exponential expression with the same base

Section 5.4 Logarithmic Functions

·  Definition of logarithm; logarithmic form and exponential form; restrictions on the bases we consider

o  Intuitively logarithm asks a question

o  logb(a) asks "What power of b is a?"

·  Definition of logarithmic functions; all have horizontal intercept (1,0)

·  Inverse Function relationship between an exponential function and the corresponding logarithmic function

·  Transformations (scaling, reflecting, shifting) of logarithmic functions

·  Common logarithm (base 10; sometimes base suppressed)

·  Natural logarithm (base e, usually written as ln(x))

·  Simple logarithmic equations: sometimes simply rewrite using exponential form will help us solve these

Section 5.5 Properties of Logarithms

·  Based on Exponential and Logarithmic Functions as Inverses (4 properties)

·  Based on Rules of Exponents (3 properties) AND Change of Base Relationship

·  Applying these properties (write as a single logarithm, expand to logarithms of "simple" expressions)

·  Using change of base to convert to base 10 or base e supported by the technology

Section 5.6 Logarithmic and Exponential Equations

·  Using Properties of Logarithms and Rules of Exponents and the facts that exponential and logarithmic functions are one to one to find exact solutions to equations

·  Using a graphing calculator to approximate a solution to an exponential or logarithmic equation

Section 5.7 Compound Interest

·  Simple Interest, Compound Interest, Continuously Compound Interest

·  Future Value A, Present Value P, Number of compoundings in one year n, Time of investment in years t

·  Nominal Annual Interest Rate expressed as a decimal r, so for example 8.347% corresponds to r = 0.08347

·  Terms for compounding frequencies: annually, semiannually, quarterly, monthly, weekly, daily

·  Solving for various parameters given values for the others: Solving for A, r, t, P; Word problems

·  Calculating and defining Effective Rate – comparing investments

·  Doubling Time (how long to double?) and generalize -- how long will it take to grow to a given size?