1 C ollege A lgebra Inverse Functions ; Exponential and Logarithmic Functions (Chapter4) L:17 1 University of Palestine IT-College

2 Chapter 4 Inverse Functions ;Exponential and Logarithmic Functions

3 Objectives: 1.Operations on Functions 2. The Difference Quotient. 3.Composition of Functions section 4.1 Function Operations and Composition

4 Operations on Functions Given two functions f and g, then for all values of x for which both f(x) and g(x) are defined, the functions f + g, f  g, fg, and are defined as follows :  (f + g)(x) = f(x) + g(x) Sum  (f  g)(x) = f(x)  g(x) Difference  (fg)(x) = f(x)  g(x) Product  Quotient

5 Example Let f(x) = x + 1 and g(x) = 2x 2  3. Find (f + g)(3) Find (fg)(  2) Solution: Since f(3) = 4 and g(3) = 15, (f + g)(3) = f(3) + g(3) (f + g)(x) = f(x) + g(x) = = 19 Since f(  2) =  1 and g(  2) = 5, (fg)(  2) = f(  2)  g(  2) (fg)(x) = f(x)  g(x) =  1  5 =  5

6 Domains For functions f and g, the domain of (f + g),( f  g), and (f.g) include all real numbers in the intersection of the domains of f and g, while the domain of includes those real numbers in the intersection of the domains of f and g for which g(x)  0. Example: Let f(x) = 3x  5 and g(x) = Find (f  g)(x) Find Give the domains of each function.

7 Domains continued Solutions: (f  g)x = f(x) – g(x) = In part (a), the domain of f is the set of all real numbers ( , ), and the domain of g, since includes just the real numbers to make 3x  2 nonnegative. That is, so.

8 Domains continued The domain of g is. The domain of f  g is the intersection of the domains of f and g, which is. The domain of includes those real numbers in the intersection above for which ; that is, the domain of is.

9 Example: Define the functions f and g as follows: Find each of the following and determine the domain of the resulting function. a.) (f + g)(x)= f(x) + g(x)

10 b.) (f - g)(x)= f(x) - g(x) c.) ( )(x)= f(x)g(x)

11 d.) We must exclude x = - 4 and x = 4 from the domain since g(x) = 0 when x = 4 or - 4.

12 The Difference Quotient Suppose the point P lies on the graph of y = f(x), and h is a positive number. If we let (x, f(x)) denote the coordinates of P and (x + h, f(x + h)) denote the coordinates of Q, then the line joining P and Q has slope This expression is called the difference quotient.

13 Example Let f(x) = x 2 + 4x. Find the difference quotient and simplify the expression. Solution: Step 1 Find f(x + h) Replace x in f(x) with x + h. f(x + h) = (x + h) 2 + 4(x + h) Step 2 Find f(x + h)  f(x) f(x + h)  f(x) = [ (x + h) 2 + 4(x + h)]  (x 2 + 4x) Substitute. = x 2 + 2xh + h 2 + 4(x + h)  (x 2 + 4x) Square x + h. = x 2 + 2xh + h 2 + 4x + 4h  x 2  4x = 2xh + h 2 + 4h

14 Example continued Step 3 Find the difference quotient.

15 Composition of Functions If f and g are functions, then the composite functions, or composition, of g and f is defined by The domain of is the set of all numbers x in the domain of f such that f(x) is in the domain of g.

16 Example Let f(x) = 3x  1 and g(x) = x + 5. Find each composition. Solution: First find g(2). Since g(x) = x + 5, g(2) = = 7. Now find = f[g(2)] = f(7): f[g(2)] = f(7) = 3(7)  1 = 20.

17 Example continued Solution: First find f(2). Since f(x) = 3x  1 f(2) = (3(2)  1) = 5 Now find = g[f(2)] = g(5) g[f(2)] = g(5) = = 10.

18 Finding Composite Functions Example: Let f(x) = x 2 and g(x) = x  2. Find and. Solution:

19 Composite Functions The previous example shows it is not always true that In fact, the composite functions are equal only for a special class of functions.

20 Finding Functions That Form a Given Composite Example: Find functions f and g such that Solution: Note the repeated quantity x If we choose g(x) = x and f(x) = x 3  2x + 7 then

21 Example: Given the functions f and g, find the domain of (f o g(.  The domain of (f o g) consists of those x in the domain of g, thus, x = - 2 is not in the domain of the composite function.  Furthermore, the domain of f requires that So:

22 Example: Given the functions f and g, find f g. 

23 section Exponential Functions Objectives After completing this section, you should be able to: 1.Evaluate an exponential function. 2.Graph exponential functions 3.Solve exponential equations. 4.Calculate compound interest problems..

24 Definition of Exponential Function Here are some examples of exponential functions. f (x) = 2 x g(x) = 10 x h(x) = 3 x+1 Base is 2.Base is 10.Base is 3. The function f defined by where b >0, b ≠1 and the exponent x is any real number, is called an exponential function.

25 Properties of Exponents If any real number a > 0, a  1, the following statements are true. –a x is a unique real number for all real numbers x. –a b = a c if and only if b = c. –If a > 1 and m < n, then a m < a n. –If 0 a n.

26 Evaluating an Exponential Expression If f(x) = 3 x, find each of the following. a) f(  1)b) f(3)c) f(3/2)d) f(5.01) Solutions: a) b) c) d)

27 Exponential Function

28 Exponential Function continued

29 Characteristics of the Graph of f(x) = a x 1. The points (0, 1), and (1, a) are on the graph. 2.If a > 1, then f is an increasing function; if 0 < a < 1, then f is a decreasing function. 3.The domain is ( ,  ), and the range is (0,  ).

30 Example Graph f(x) = 6 x. y-intercept = 1 x-axis = horizontal asymptote Domain: ( ,  ) Range (0,  ) 11 f(x)f(x)x

Transformations Involving Exponential Functions Shifts the graph of f (x) = b x upward c units if c > 0. Shifts the graph of f (x) = b x downward c units if c < 0. g(x) = -b x + cVertical translation Reflects the graph of f (x) = b x about the x-axis. Reflects the graph of f (x) = b x about the y-axis. g(x) = -b x g(x) = b -x Reflecting Multiplying y-coordintates of f (x) = b x by c, Stretches the graph of f (x) = b x if c > 1. Shrinks the graph of f (x) = b x if 0 < c < 1. g(x) = c b x Vertical stretching or shrinking Shifts the graph of f (x) = b x to the left c units if c > 0. Shifts the graph of f (x) = b x to the right c units if c < 0. g(x) = b x+c Horizontal translation DescriptionEquationTransformation

32 Graphing Reflections and Translations Graph each function. a) f(x) =  3 x b) f(x) = 3 x + 2 c) f(x) = 3 x + 2

33 Solution f(x) =  3 x Reflected across the x-axis. Domain: ( ,  ) Range: ( , 0)

34 f(x) = 3 x + 2 The graph of f(x) = 3 x translated 2 units to the left. Solution

35 f(x) = 3 x + 2 The graph of f(x) = 3 x translated 2 units up. Solution

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38 Definition and Graph of the Natural Exponential Function

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The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to More accurately, The number e is called the natural base. The function f (x) = e x is called the natural exponential function. f (x) = e x f (x) = 2 x f (x) = 3 x (0, 1) (1, 2) (1, e) (1, 3)

41 Solving Exponential Equations, where x is in the exponent, BUT the bases DO NOT MATCH. Step 1: Isolate the exponential expression. Get your exponential expression on one side everything outside of the exponential expression on the other side of your equation. Step 2: Take the natural log of both sides. The inverse operation of an exponential expression is a log. Make sure that you do the same thing to both sides of your equation to keep them equal to each other. Step 3: Use the properties of logs to pull the x out of the exponent. Step 4: Solve for x. Now that the variable is out of the exponent, solve for the variable using inverse operations to complete the problem.

42 Example 1: Solve the exponential equation Round your answer to two decimal places. Step 1: Isolate the exponential expression.Isolate the exponential expression. This is already done for us in this problem. Step 2: Take the natural log of both sides.Take the natural log of both sides. Step 3: Use the properties of logs to pull the x out of the exponent.Use the properties of logs to pull the x out of the exponent.

43 Step 4: Solve for x.Solve for x.

44 Example 2: Solve the exponential equation Round your answer to two decimal places. Step 1: Isolate the exponential expression.Isolate the exponential expression. Step 2: Take the natural log of both sides.Take the natural log of both sides. Step 3: Use the properties of logs to pull the x out of the exponent.Use the properties of logs to pull the x out of the exponent.

45 Step 4: Solve for x.Solve for x.

46 Example 3: Solve the exponential equation Round your answer to two decimal places. Step 1: Isolate the exponential expression.Isolate the exponential expression. Step 2: Take the natural log of both sides.Take the natural log of both sides. Step 3: Use the properties of logs to pull the x out of the exponent.Use the properties of logs to pull the x out of the exponent.

47 Step 4: Solve for x.Solve for x.

48 Exponential Equations Solve

49 Another Example Solve 3 x + 1 = 27 x  3

50 Compound Amount and Interest Compound interest means that at the end of each interest period the interest earned for that period is added to the previous principle (the invested amount) so that, it too, earns interest over the next interest period. In other words this is an accumulative interest. Compound Interest Formula where, S = compound or accumulated amount P = Principal (starting value ( r = nominal rate (annual % rate ( n = the number of compound periods per year t = number of years

51 Example 6: Find the a) compound amount AND b) the compound interest for the given investment and rate. $15000 for 14 years at an annual rate of 5% compounded monthly. P = r = 5% =.05 t = 14 n = monthly = 12 times a year So the compound AMOUNT would be $ &compound interest is $ compound interest =Compound amount - principle

52 Compound Continuously Formula If P dollars are deposited at a rate of interest r compounded continuously for t years, the compounded amount in dollars on deposit is A = Pe rt. where, S = compound or accumulated amount P = Principal (starting value) r = nominal rate (annual % rate) t = number of years Compounded continuously means that it is compounded at every instant of time

53 Example Suppose $2000 is deposited in an account paying 2% interest compounded continuously for 3 yr. Find the total amount on deposit at the end of the 3 yr. Solution:

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