Exponential and Logarithmic Functions Chapter 9 Exponential and Logarithmic Functions

The Algebra of Functions; Composite Functions § 9.1 The Algebra of Functions; Composite Functions

Operations on Functions It is possible to add, subtract, multiply, and divide functions. The results of these operations will also be functions (assuming we don’t divide by zero).

Operations on Functions Algebra of Functions Let f and g be functions. New functions from f and g are defined as follows: Sum (f + g)(x) = f(x) + g(x) Difference (f – g)(x) = f(x) – g(x) Product (f · g)(x) = f(x) · g(x) Quotient

Operations on Functions Example: If f(x) = 4x + 3 and g(x) = x2, then find each of the following (f + g)(x) 4x + 3 + x2 = x2 + 4x + 3 (f – g)(x) 4x + 3 – x2 = -x2 + 4x + 3 (f · g)(x) (4x + 3)x2 = 4x3 + 3x2

Operations on Functions Example: If f(x) = 4x + 3 and g(x) = x2, then find x  0

Function Composition We can also combine functions through a function composition. A function composition uses the output from the first function as the input to the second function. Composition of a Function The composition of function f and g is This means the value of x is first substituted into the function g. Then the value that results from the function g is input into the function f.

Function Composition Notice, that with function composition, we actually activate the functions from right to left in the notation. The function named on the right side of the composition notation is the one we substitute the value for the variable into first.

Function Composition Example: If f(x) = 4x + 3 and g(x) = x2, then find f(g(x)) = 4(x2) + 3 = 4x2 + 3 g(f(x)) = (4x + 3)2 = 16x2 + 24x + 9 Notice the results are different with a different order.

Function Composition Example: If H(x) = x3 + 3, name two functions whose composition will result in H(x). Note: There may be more than one way to select the two functions. Answers are not necessarily unique. Let f(x) = x + 3, and g(x) = x3

Exponential Functions § 9.3 Exponential Functions

Exponential Expressions We have previously worked with exponential expressions, where the exponent was a rational number The expression bx can actually be defined for all real numbers, x, including irrational numbers. However, the proof of this would have to wait until a higher level math course.

Exponential Functions A function of the form f(x) = bx is called an exponential function if b > 0, b is not 1, and x is a real number.

Exponential Functions We can graph exponential functions of the form f(x) = 3x, g(x) = 5x or h(x) = (½)x by substituting in values for x, and finding the corresponding function values to get ordered pairs. We would find all graphs satisfy the following properties: 1-to-1 function y-intercept (0, 1) no x-intercept domain is (-, ) range is (0, )

Graphs of Exponential Functions y We would find a pattern in the graphs of all the exponential functions of the type bx, where b > 1.

Graphs of Exponential Functions y We would find a pattern in the graphs of all the exponential functions of the type bx, where 0 < b < 1.

Graphs of Exponential Functions y (h, 1) We would find a pattern in the graphs of all the exponential functions of the type bx-h, where b > 1. The graph has the same shape as the graph for bx, except it is shifted to the right h units.

Graphs of Exponential Functions y (-h, 1) We would find a pattern in the graphs of all the exponential functions of the type bx+h, where b > 1. The graph has the same shape as the graph for bx, except it is shifted to the left h units.

Uniqueness of bx Uniqueness of bx Let b > 0 and b  1. Then bx = by is equivalent to x = y. Example: Solve 6x = 36 6x = 62 x = 2

Solving Exponential Functions Example: Solve 92x+1 = 81 92x+1 = 92 2x + 1 = 2 2x = 1 x = ½

Solving Exponential Functions Example: Solve 3-3 = 32x –3 = 2x

Solving Exponential Functions Example: Solve 43x-6 = 322x (22)3x-6 = (25)2x (22)3x-6 = 210x 26x-12 = 210x 6x – 12 = 10x –12 = 4x x = –3

Applications of Exponential Functions Many applications use exponential functions of various types. Compound interest formulas are exponential functions used to determine the amount of money accumulated or borrowed. Exponential functions with negative exponents can be used to describe situations of decay, while those with positive exponents can be used to describe situations of growth.

Applications of Exponential Functions Example: Find the total amount invested in a savings account if $5000 was invested and earned 6% compounded monthly for 18 years. Round your answer to two decimal places. The formula that is used for calculating compound interest is where P is the initial principal invested, r is the interest rate, n is the number of times interest is compounded each year, t is the time of the investment (in years) and A is the amount of money in the account. Continued.

Applications of Exponential Functions Example continued: $

Applications of Exponential Functions Example: An accidental spill of 100 grams of radioactive material in a local stream has led to the presence of radioactive debris decaying at a rate of 5% each day. Find how much debris still remains after 30 days. The formula that would be used for this problem is where A is the amount of radioactive material to start, r is the rate of decay, t is the number of days and y is the amount of radioactive material after the time period. Continued.

Applications of Exponential Functions Example continued: (exact answer)

9.1 #'s 1-23 odd 9.3  #'s 1, 5, 17, 19, 21, 23, 29, 31, 33

Logarithmic Functions § 9.4 Logarithmic Functions

Graph of a Exponential Function If we graph an exponential function where the base > 1, we get an increasing function, as shown below. x y

Graph of a Logarithmic Function We can graph the inverse of the function, as shown below. x y This inverse function is referred to as a logarithmic function.

Logarithmic Functions Logarithmic Definition If b > 0 and b ≠ 1, then y = logb x means x = by for every x > 0 and every real number y.

Writing Exponential Functions Example: Write each of the following as an exponential equation. log4 16 = 2 4² = 16 log8 ⅛ = –1 8–1 = ⅛ c) log3 = ½

Writing Logarithmic Functions Example: Write each of the following as a logarithmic equation. a) 54 = 625 log5 625 = 4 b) 2–3 = ⅛ log2 ⅛ = –3 c) 41/3 = log4 = ⅓

Values of Logarithmic Expressions Example: Find the value of each of the following logarithmic expressions. a) log2 32 Since 25 = 32, then log2 32 = 5 b) log5 Since 5–2 = , then log5 = –2 c) log4 2 Since 4½ = 2, then log4 2= ½

Solving Logarithmic Equations Example: Solve log3 1 = x for x. First we rewrite the equation as an exponential equation. 3x = 1 Since 30 = 1, then x = 0.

Solving Logarithmic Equations Example: Solve logx 81 = 4 for x. First we rewrite the equation as an exponential equation. x4 = 81 Since 34 = 81, then x = 3.

Solving Logarithmic Equations Example: Solve log6 x = 2 for x. First we rewrite the equation as an exponential equation. 62 = x Since 62 = 36, then x = 36.

Properties of Logarithms 1) logb 1 = 0 2) logb bx = x 3) b = x logb x

Properties of Logarithms Example: Simplify each of the following expressions 1) log4 46 From Property 2, log4 46 = 6. 2) 7 log7 –3 From Property 3, 7 = –3. log7 –3

Logarithmic Functions If x is a positive real number, b is a constant positive real number, and b is not 1, then a logarithmic function is a function that can be defined by f(x) = logb x The domain of f is the set of positive real numbers, and the range of f is the set of real numbers. A logarithmic function is an inverse function of an exponential function.

Graphs of Logarithmic Functions To graph a logarithmic function, we first write the equation in exponential notation. Then we find ordered pairs that satisfy the equation and plot their corresponding points.

Graphs of Logarithmic Functions Example: Graph y = log2 x. x y Write the exponential form 2y = x. Then choose y-values and find corresponding x-values. x y 4 2 2 1 1 ½ -1 ¼ -2

Graphs of Logarithmic Functions Example: Graph y = log½ x. x y Write the exponential form (½)y = x. Then choose y-values and find corresponding x-values. x y ¼ 2 ½ 1 1 2 -1 4 -2

Logarithmic Functions In general, from these two previous examples, we would find that for the logarithmic function f(x) = logb x, b > 0, b ≠ 1, The function is a 1-to-1 function. has a domain of (0, ∞). has a range (–∞, ∞). has an x-intercept of (1, 0). has no y-intercept.

The Parabola and the Circle § 10.1 The Parabola and the Circle

Conic Sections Conic sections derive their name because each conic section is the intersection of a right circular cone and a plane. Circle Ellipse Parabola Hyperbola

The Parabola Just as y = a(x – h)2 + k is the equation of a parabola that opens upward or downward, x = a(y – k)2 + h is the equation of a parabola that opens to the right or to the left. y = a(x – h)2 + k x = a(y – k)2 + h x y x y x y x y a > 0 (h, k) (h, k) (h, k) y = k a < 0 y = k (h, k) a < 0 a > 0 x = h x = h

The Parabola Example: Graph the parabola x = (y – 4)2 + 1. a > 0, so the parabola opens to the right. The vertex of the parabola is (1, 4). The axis of symmetry is y = 4.

The Parabola Example continued: x y 2 The table shows ordered pairs of the solutions of x = (y – 4)2 + 1. x y 1 4 y = 4 2 3 2 5 17 17 8

The Parabola Example: Graph the parabola y = x2 + 12x + 25. Complete the square on x to write the equation in standard form. y – 25 = x2 + 12x Subtract 25 from both sides. The coefficient of x is 12. The square of half of 12 is 62 = 36. y – 25 + 36 = x2 + 12x + 36 Add 36 to both sides.

The Parabola Example continued: y + 11 = (x + 6)2 Simplify the left side and factor the right side. y = (x + 6)2 – 11 Subtract 11 from both sides. a > 0, so the parabola opens upward. The vertex of the parabola is (– 6, – 11). The axis of symmetry is x = – 6.

The Parabola Example continued: x y 3 y = x2 + 12x + 25

The Distance Formula Distance Formula The distance d between any two points (x1, y1) and (x2, y2) is given by y x (x1, y1) (x2, y2) d a = x2 – x1 b = y2 – y1

The Distance Formula Example: Find the distance between (– 6, – 6) and (– 5, – 2).

The Midpoint The midpoint of a line segment is the point located exactly halfway between the two endpoints of the line segment. Midpoint Formula The midpoint of the line segment whose endpoints are (x1, y1) and (x2, y2) is the point with the coordinates

The Midpoint Example: Find the midpoint of the line segment that joins points P(0, 8) and Q(4, – 6).

The Cirlce A circle is the set of all points in a plane that are the same distance from a fixed point called the center. The distance is called the radius. Circle The graph of (x – h)2 + (y – k)2 = r2 is a circle with center (h, k) and radius r. y x (h, k) r

The Circle Example: Graph (x – 3)2 + y2 = 9. The equation can be written as (x – 3)2 + (y – 0)2 = 32. h = 3, k = 0, and r = 3. x y r = 3 (3, 0)

The Circle Example: Find the equation of the circle with center (– 7, 6) and radius 2. h = – 7, k = 6, and r = 2. (x – h)2 + (y – k)2 = r2. The equation can be written as [x – (– 7)2] + (y – 6)2 = 22. (x + 7)2 + (y – 6)2 = 4. Simplify.