ࡱ> uw tR@ 3bjbjFF ",,+HHHH\6J,1vnK |V|V|V|Vkkk$R֢kYk@kkk֢|V|Vlllk"|V|VlkllpXX|VbK O HkȜ(01hal.PXXkklkkkkk֢֢d.0l0CHAPTER 4: EXPONENTIAL AND LOGARITHMIC FUNCTIONS INVERSE FUNCTIONS INVERSES Inverse Relation Interchanging the first and second coordinates of each ordered pair in a relation produces the inverse relation. If a relation is defined by an equation, interchanging the variables produces an equation of the inverse relation. Example: Find the inverse of the relation  EMBED Equation.DSMT4  Solution:  EMBED Equation.DSMT4  Notice that the pairs in the inverse are reflected across the line  EMBED Equation.DSMT4   Example: Find an equation of the inverse relation  EMBED Equation.DSMT4  Solution:  EMBED Equation.DSMT4   One-to-One Functions A function  EMBED Equation.DSMT4  is one-to-one if different inputs have different outputsthat is, if  EMBED Equation.DSMT4  then  EMBED Equation.DSMT4  Or a function  EMBED Equation.DSMT4  is one-to-one if when the outputs are the same, the inputs are the samethat is if  EMBED Equation.DSMT4  then  EMBED Equation.DSMT4  Properties of One-to-One Functions and Inverses If a function is one-to-one, then its inverse is a function. The domain of a one-to-one function  EMBED Equation.DSMT4  is the range of the inverse  EMBED Equation.DSMT4  The range of a one-to-one function  EMBED Equation.DSMT4  is the domain of the inverse  EMBED Equation.DSMT4  A function that is increasing over its domain or is decreasing over its domain is a one-to-one function. Horizontal Line Test If it is possible for a horizontal line to intersect the graph of a function more than once, then the function is not one-to-one and its inverse is not a function. Finding Formulas for Inverses Obtaining a Formula for an Inverse If a function  EMBED Equation.DSMT4  is one-to-one, a formula for its inverse can generally be found as follows: Replace  EMBED Equation.DSMT4  with  EMBED Equation.DSMT4  Interchange  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Solve for  EMBED Equation.DSMT4  Replace  EMBED Equation.DSMT4  with  EMBED Equation.DSMT4  Example: If the function is one-to-one, find a formula for the inverse:  EMBED Equation.DSMT4  Solution: First we graph the function to see if it will pass the horizontal line test. As you can see from the graph below, it will, so we have a one-to-one function.  Now we can use the steps above to find the inverse function. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  The graph of  EMBED Equation.DSMT4  is a reflection of the graph of  EMBED Equation.DSMT4  across the line  EMBED Equation.DSMT4  Inverse Functions and Composition If a function  EMBED Equation.DSMT4  is one-to-one, then  EMBED Equation.DSMT4  is the unique function such that each of the following holds:  EMBED Equation.DSMT4  Restricting a Domain If the inverse of a function is not a function, we can restrict the domain so that the inverse is a function. Example: Consider  EMBED Equation.DSMT4  If we try to find a formula for the inverse, we have  EMBED Equation.DSMT4  This is not the equation of a function. We can, however, only consider inputs from  EMBED Equation.DSMT4  This will yield an inverse that is a function. EXPONENTIAL FUNCTIONS AND GRAPHS Graphing Exponential Functions Exponential Function The function  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is a real number,  EMBED Equation.DSMT4  is called the exponential function, base  EMBED Equation.DSMT4  Example: Graph  EMBED Equation.DSMT4   EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 01 EMBED Equation.DSMT4 -1 EMBED Equation.DSMT4  EMBED Equation.DSMT4 -2 EMBED Equation.DSMT4  EMBED Equation.DSMT4 -3 EMBED Equation.DSMT4  EMBED Equation.DSMT4 12 EMBED Equation.DSMT4 24 EMBED Equation.DSMT4 38 EMBED Equation.DSMT4  Applications Compound Interest  EMBED Equation.DSMT4  The number e Using the formula  EMBED Equation.DSMT4 , we suppose that $1 is invested at 100% interest for 1 year. This gives us  EMBED Equation.DSMT4  You will find that as  EMBED Equation.DSMT4   EMBED Equation.DSMT4  So  EMBED Equation.DSMT4  Graph of  EMBED Equation.DSMT4   LOGARITHMIC FUNCTIONS AND GRAPHS WARM-UP Graph the exponential function  EMBED Equation.DSMT4  and the line  EMBED Equation.DSMT4  on the same set of axes.  EMBED Equation.DSMT4    EMBED Equation.DSMT4   EMBED Equation.DSMT4 0123-1-2 EMBED Equation.DSMT4 1248 EMBED Equation.DSMT4  EMBED Equation.DSMT4   EMBED Equation.DSMT4 0123-1-2 EMBED Equation.DSMT4 0123-1-2  EMBED Equation.DSMT4 1248 EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 0123-1-2Recall that if you have a one-to-one function, its inverse is a function. Is  EMBED Equation.DSMT4  a one-to-one function? _______ Why? Recall that a function of the form  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 is called an exponential function. Also recall that when we find an inverse algebraically we Change  EMBED Equation.DSMT4  to EMBED Equation.DSMT4 . Swap  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Solve for  EMBED Equation.DSMT4 . Change  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4 . Lets try finding the inverse of  EMBED Equation.DSMT4  algebraically.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  We havent done this before! This is where logarithms come in to play. Always remember A LOGARITHM IS AN EXPONENT!!! Also remember that the word is in math language means equals.  Definition of Logarithm: If  EMBED Equation.DSMT4  then  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  means the log (or logarithm) of x to base b. In plain words we say the power to which we raise b to get x when we see  EMBED Equation.DSMT4 . Lets practice: Write each of the following logarithmic equations as an exponential equation.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Write each of the following exponential equations as an exponential equation.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Natural Logarithms Logarithms, base e, are called natural logarithms. The abbreviation ln is used for natural logarithms.  EMBED Equation.DSMT4  Changing Logarithmic Bases The Change-of-Base Formula For any logarithmic bases a and b, and any positive number m,  EMBED Equation.DSMT4  PROPERTIES OF LOGARITHMIC FUNCTIONS Logarithms of Products The Product Rule For any positive numbers M and N and any logarithmic base a,  EMBED Equation.DSMT4  The logarithm of a product is the sum of the logarithms of the factors. Proof: Let  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  So we have,  EMBED Equation.DSMT4  Now,  EMBED Equation.DSMT4  This gives us  EMBED Equation.DSMT4  Substituting, we have  EMBED Equation.DSMT4  The Power Rule For any positive number M, any logarithmic base a, and any real number p,  EMBED Equation.DSMT4  The logarithm of a power of M is the exponent times the logarithm of M. Proof: Let  EMBED Equation.DSMT4  So we have,  EMBED Equation.DSMT4  Now,  EMBED Equation.DSMT4  This gives us  EMBED Equation.DSMT4  Substituting, we have  EMBED Equation.DSMT4  The Quotient Rule For any positive numbers M and N and any logarithmic base a,  EMBED Equation.DSMT4  The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator. Proof:  EMBED Equation.DSMT4  Properties of Logarithms: If  EMBED Equation.DSMT4   EMBED Equation.DSMT4  SOLVING EXPONENTIAL AND LOGARITHMIC FUNCTIONS Equations with variables in the exponents are called exponential equations One way to solve these equations is to manipulate each side so that each side is a power of the same number. Example:  EMBED Equation.DSMT4  can be written as  EMBED Equation.DSMT4  Then we just set the quantities in the exponents equal to each other and solve.  EMBED Equation.DSMT4  Base-Exponent Property For any  EMBED Equation.DSMT4  Example: Solve  EMBED Equation.DSMT4  Solution:  EMBED Equation.DSMT4  Property of Logarithmic Equality For any  EMBED Equation.DSMT4  Solving Logarithmic Equations Equations containing variables in logarithmic expressions are called logarithmic equations. To solve, we try to obtain a single logarithmic expression on one side of the equation and then write an equivalent exponential equation. Example: Solve  EMBED Equation.DSMT4  Solution:  EMBED Equation.DSMT4  APPLICATIONS AND MODELS: GROWTH AND DECAY, AND COMPOUND INTEREST Population Growth The function  EMBED Equation.DSMT4  is a model of many kinds of population growth  EMBED Equation.DSMT4  is the population at time 0,  EMBED Equation.DSMT4  is the population after time  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  is called the exponential growth rate. Interest Compounded Continuously The function  EMBED Equation.DSMT4  is also a model for compound interest  EMBED Equation.DSMT4  is the initial investment,  EMBED Equation.DSMT4  is the amount of money in the account after  EMBED Equation.DSMT4  years, and  EMBED Equation.DSMT4  is the interest rate compounded continuously Growth Rate and Doubling Time The growth rate  EMBED Equation.DSMT4  and the doubling time  EMBED Equation.DSMT4  are related by  EMBED Equation.DSMT4  Proof: If we substitute  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4  we have,  EMBED Equation.DSMT4  Models of Limited Growth  EMBED Equation.DSMT4  is a logistic function which increases toward a limiting value  EMBED Equation.DSMT4  Therefore, the line  EMBED Equation.DSMT4  is the horizontal asymptote of the graph of  EMBED Equation.DSMT4  Used in situations where there are factors that prevent a population from exceeding some limiting value Exponential Decay The function  EMBED Equation.DSMT4  is an effective model of the decline or decay of a population or substance  EMBED Equation.DSMT4  is the amount of the substance at time 0,  EMBED Equation.DSMT4  is the amount still radioactive after time  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  is called the decay rate. 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