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Math 150 – Fall 2015 Section 5C 1 of 5 Section 5C – Exponential Functions 3 Example 1. What is 16 2 ? √ Question. What is 2 2 ? Theorem. To evaluate ab , when b is irrational (so b is not a fraction of integers), we p approximate it by choosing rational numbers pq that are close to b and evaluating a q . We can find rational numbers pq as close to b as we want. As we let our fraction pq p approach b, the numbers a q are also approaching a number, we call this number ab . √ Example 2. Compute 2x for a sequence of x’s getting close to 2. Properties of Exponents: For any positive real number a, and any real numbers x and y, the following properties hold: 1. ax ay = ax+y 2. ax ay = ax−y 3. a−x = 1 ax 4. a0 = 1 for a 6= 0 x 5. (ab) = ax bx x x 6. ab = abx y 7. (ax ) = axy Definition of Exponential Functions Definition. The function f (x) = ax , with 0 < a and x any real number, is called the exponential function with base a x Example 3. Fill in the table below for the functions f (x) = 2x and f (x) = 12 . Use the points to graph the function. x -3 1 x 2 2x -2 -1 0 1 2 3 . Math 150 – Fall 2015 Section 5C 2 of 5 Theorem. Consider the exponential function, f (x) = ax . 1. If a > 1, then the graph of ax looks like the graph of 2x . The higher the value of a the steeper the graph will be. 2. If 0 < a < 1, then the graph of ax looks like the graph of value of a, the steeper the graph will be. a x . 2 The smaller the 3. If a = 1, then the graph of ax is the horizontal line y = 1. Properties for f (x) = ax : You need to know these properties! 1. The domain of f (x) = ax is all real numbers. 2. The range of f (x) = ax , if a 6= 1, is all positive real numbers. If a = 1, then the range is the set consisting of the number 1 only. Note: ax can never equal 0. 3. If a > 1, then as x → ∞, f (x) = ax → ∞, and as x → −∞, f (x) = ax → 0. 4. If 0 < a < 1, then as x → ∞, f (x) = ax → 0, and as x → −∞, f (x) = ax → ∞. 5. If 1 < a and x < y, then ax < ay . Therefore, f (x) = ax is an increasing function if a > 1. 6. If 0 < a < 1 and x < y, then ax > ay . Therefore, f (x) = ax is an decreasing function if 0 < a < 1. Math 150 – Fall 2015 Section 5C Example 4. Which number is bigger: 1 6 3 or 3 of 5 1 5 3 ? Example 5. What does the plot of 2 · 2x look like compared to the plot of 2x ? Consider the graphs below of y = 1.5x , y = 2x , y = 2.5x , y = 3x , and y = 3.5x for −1 ≤ x ≤ 1. Also, compare the graphs of y = a0.25 , y = a0.5 , and y = a.75 . Theorem. In general, we have the following: If a < b, and x > 0, then ax < bx . If a < b, and x < 0, then ax > bx . Example 6. Which number is larger: √ √ (a) 2− 2 or 3− 2 π π (b) 23 or 73 (c) √ 5 − 7 2 or √ 13 − 7 11 Math 150 – Fall 2015 Section 5C 4 of 5 The Natural Exponential Function Definition. The natural exponential function is the function f (x) = ex , where e 1 n is the limit of 1 + n as n goes to infinity. Note. The value of e can be approximated by substituting larger and larger numbers n for n in the expression 1 + n1 . A closer approximation is e ≈ 2.71828182845905 Then 2 < e < 3 so the graph of ex is between the graphs of 2x and 3x . Example 7. Place the following numbers in increasing order: 2.85 , 2.33 , e4 , 35 , e5 , 2.54 . Theorem. To evaluate ex , we can use the following limit x n ex = lim 1 + n→∞ n x Example 8. Use the formula ex = limn→∞ 1 + nx to approximate e3 (You need a calculator) n n 1 + n3 1 10 100 1000 10,000 100,000,000 Math 150 – Fall 2015 Section 5C 5 of 5 Example 9. For the following functions, find the domain, range, x-intercepts, and y-intercepts. Also, describe the end behavior of the functions and the intervals where the function is increasing and decreasing. (a) f (x) = −3ex−1 − 2 (b) g(x) = 2 1 x+1 3 − 18 (c) h(x) = −2(4)−x−2 + 4