Inverse Functions Definition 1. The exponential function f with base a is denoted by f(x) = a x where a > 0, a 1, and x is any real number. Example 1. In the same coordinate plane, sketch the graph of each function. 1. f(x) = 2 x 2. g(x) = 4 x Example 2. In the same coordinate plane, sketch the graph of each function. 1. f(x) = 2 x 2. g(x) = 4 x Chapter 2: 2.2 Exponential Functions 1
Chapter 2: 2.2 Exponential Functions 2
Properties of Exponential Functions f(x) = a x, a > 0, a 1 1. The domain of f is the set of real numbers. 2. The range of f is the set of positive real numbers. 3. The y-intercept is at point (0, 1), that is, f(0) = 1. 4. f(1) = a 5. If 0 < a < 1, as the value of x increases, the value of y decreases. 6. If a > 1, as the value of x increases, the value of y increases. 7. The function f is one-to-one, that is, a x = a y if and only if x = y. 8. The graph of f has a horizontal asymptote y = 0 or the x- axis. Chapter 2: 2.2 Exponential Functions 3
Laws of Exponents Let a and b be positive real numbers, a 1, b 1, and let x and y be real numbers. 1. a x a y = a x+y 2. (a x ) y = a xy 3. (ab) x = a x b x ( ) x a = ax 4. b b x 5. ax a y = ax y Chapter 2: 2.2 Exponential Functions 4
Example 3. Solve 5 2x+1 = 25. Example 4. Solve 2 3x = 16 1 x. Chapter 2: 2.2 Exponential Functions 5
Solving Exponential Inequalities 1. Write the inequality in the standard form. One side must be zero and the other side is a single exponential expression which we denote by f(x). 2. Find the critical numbers. These are the zeros of f(x). 3. Divide the number line into intervals according to the critical numbers obtained in Step 2. 4. Choose a test value in each interval in Step 3, and construct a table. Substitute the test value to f(x) and determine the sign of the resulting answer. The sign of this answer (positive or negative) will be the sign of the entire interval. You can check using a different number from the same interval if you want to verify your answer. Chapter 2: 2.2 Exponential Functions 6
5. Use the table in step 4 to determine which intervals satisfy the inequality. If the inequality is of the form f(x) < 0 or f(x) 0 then all of the intervals with the negative sign are solutions. In addition, the zeros of f(x) are part of the solution if f(x) 0. On the other hand, if the inequality is of the form f(x) > 0 or f(x) 0 then all of the intervals with the positive sign are solutions. In addition, the zeros of f(x) are part of the solution if f(x) 0. Chapter 2: 2.2 Exponential Functions 7
Example 5. Solve 2 x+1 > 32. Example 6. Solve 4 3x+2 64. Chapter 2: 2.2 Exponential Functions 8
Doubling Time Growth Model The doubling time growth model is defined as P (t) = P 0 2 t/d where P (t) = population at time t P 0 = population at time t = 0 d = doubling time Chapter 2: 2.2 Exponential Functions 9
Example 7. The bacterium Escherichia coli (E. coli) is found naturally in the intestines of many mammals. In a particular laboratory experiment, the doubling time for E. coli is found to be 30 minutes. If the experiment starts with a population of 1,000 E. coli and there is no change in the doubling time, how many bacteria will be present: (a) in 15 minutes? (b) in 4 hours? Chapter 2: 2.2 Exponential Functions 10
Half-life Decay Model The half-life decay model is defined as ) t/h A(t) = A 0 ( 1 2 where P (t) = amount at time t A 0 = amount at time t = 0 h = half-life Chapter 2: 2.2 Exponential Functions 11
Example 8. You discovered a new radioactive isotope. It s half life is 1.23 years. If you start with a sample of 45 grams, how much will be left (a) in 10 months? (b) in 7.6 years? Chapter 2: 2.2 Exponential Functions 12
Compound Interest The compound interest equation is F = P ( 1 + j m ) tm where P = principal or initial investment F = accumulated value or final amount of the investment j= the annual interest rate m = the number of times interest is paid or compounded each year t= the number of years Chapter 2: 2.2 Exponential Functions 13
Example 9. If P10,000 is invested in an account paying 10% compounded monthly, how much will be in the account at the end of 10 years? Compute the answer to the nearest cent. Chapter 2: 2.2 Exponential Functions 14
Definition 2. For x a real number, the equation f(x) = e x defines the exponential function with base e. Continuously Compounded Interest The continuously compounded interest equation is where F = P e rt P = principal or initial investment F = accumulated value or final amount of the investment r = the annual interest rate t= the number of years Chapter 2: 2.2 Exponential Functions 15
Example 10. If P10,000 is invested in an account paying 10% compounded continuously, how much will be in the account at the end of 10 years? Compute the answer to the nearest cent. Chapter 2: 2.2 Exponential Functions 16
Exercises Construct a table of values for the function and sketch the graph of the function. Identify any asymptote of the graph. Verify your answer using any graphing utility. 1. f(x) = 2. f(x) = ( 3 2 ( 3 2 ) x ) x 6. f(x) = 4 x 3 + 1 7. f(x) = 2 x2 8. f(x) = e x/2 3. f(x) = 6 x 4. f(x) = 2 x 1 9. h(t) = 3e 0.2t 10. f(x) = 2e 1 x 5. f(x) = 3 x+2 Chapter 2: 2.2 Exponential Functions 17
Solve each equation. 1. 4 2x+3 = 1 2. 5 3 2x = 5 x 7. 4 2 1 x = 1 3. 3 1 2x = 243 4. e 2x = e x 8. 64 x+3 4 x 2 = 8 9. 243 2x 1 = 27 3x+2 5. 5 2+2x = 1 5 6. e3x+2 e x = 1 10. ( 1 ) 2x 2 3 27 = ( 1 81 ) 2x 2 Chapter 2: 2.2 Exponential Functions 18
Solve each inequality. 1. 3 2x 4 < 9 2. 2 3x+8 8 3. 4 x+1 32 4. 5 x 4 625 5. 32 x 1 < 4 Chapter 2: 2.2 Exponential Functions 19
Answer the following questions. 1. There are three options for investing P 50,000. The first earns 8% compounded semi-annually, the second earns 8% compounded monthly, and the third earns 8% compounded continuously. Which investment yields the highest return after 20 years. What is the difference in earnings between each investment? 2. If a certain country has a population of about 30,000,000 people and a doubling time of 19 years and if the growth continues at the same rate, find the population in: (a) 10 years (b) 30 years. Round off answers to the nearest whole number. 3. Radioactive gold 198, used in imaging the structure of the liver, has a half-life of 2.67 days. If we start with 50 milligrams of the isotope, how many milligrams will be left after: Chapter 2: 2.2 Exponential Functions 20
(a) 3 days (b) 2 weeks. 4. A certain type of bacteria increases according to the model P (t) = 100e 0.1279t where t is the time in hours. Find the amount of bacteria at (a) 0 hours (b) 5 hours (c) 10 hours 5. A computer laptop has a suggested retail price of P36,999. ( ) After t t 3 months, the laptop s value is given by V (t) = 36, 999. Find the 4 value of the laptop after (a) 6 months (b) 1 year Chapter 2: 2.2 Exponential Functions 21