Exponential Functions:

Transcription

1 Exponential Functions: An exponential function has the form f (x) = b x where b is a fixed positive number, called the base. Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

2 Exponential Functions: An exponential function has the form f (x) = b x where b is a fixed positive number, called the base. Examples: g(x) = e x, h(x) = 3 x, and k(x) = 4 x are all exponential functions. Remember: e is an irrational number that shows up in many different places; like π it shows up enough that we give it s own name. e The function f (x) = e x is very important in Calculus Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

3 In Class Work 1. (a) Is π x an exponential function? (b) Is x 2 an exponential function? 2. If f (x) = b x, where b is a fixed positive number, (a) what is f (0)? (Remember that b is a fixed number) (b) what is f (1)? (Again, remember that b is a fixed number) 3. By plotting a few points, sketch the graphs of y = f (x), where: (a) f (x) = 2 x (b) f (x) = 3 x Once you have a sketch, feel free to use graphing calc to check 4. If b > 0, what is the domain of f (x) = b x? How about the range? Remember: the domain is the set of inputs what x does it make sense to plug in; the range is the set of outputs what values of f (x) will you get out, if you plug in all possible values of x? Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

4 Solutions - In Class Work 1. (a) Is π x an exponential function? Yes: the variable is in the exponent, while the the base is a constant. (b) Is x 2 an exponential function? No: the variable is in the base, while the exponent is fixed. This type of function is called a power function. 2. If f (x) = b x, where b is any positive number, (a) what is f (0)? f (0) = b 0 = 1. Conclusion: The point (0, 1) is on every exponential function. (b) what is f (1)? f (1) = b 1 = b. Conclusion: The graph of b x goes through the point (1, b) for all b Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

5 Solutions - In Class Work 3. By plotting a few points, sketch the graphs of y = f (x), where (a) f (x) = 2 x (b) f (x) = 3 x x (a) f (x) = 2 x y x (b) f (x) = 3 x y = = = = = = = = = = = = = = 9 Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

6 Solutions - In Class Work 3. By plotting a few points, sketch the graphs of (a) y = 2 x (b) y = 3 x Note: As long as the constant b > 1, the graph of y = b x will be increasing, and concave up. Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

7 Solutions - In Class Work 4. If b > 0, what is the domain of f (x) = b x? How about the range? Remember: the domain is the set of inputs what x does it make sense to plug in; the range is the set of outputs what values of f (x) will you get out, if you plug in all possible values of x? (a) As long as b > 0, b x makes sense for all possible values of x, so the domain is all real numbers, or the interval (, ). (b) As long as b > 0, b x will also be positive. We can get any value out, by choosing an appropriate value of x. Thus the range is all positive numbers, or (0, ). Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

8 Logarithmic Functions Recall: A logarithm function with base b is denoted f (x) = log b (x) and is defined by y = log b (x) iff b y = x. Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

9 Comparison: Exponential & Log Fns; Cube and Cube Root Fns Definition of Logarithm: y = log b (x) b y = x Definition of Cube Root: y = 3 x y 3 = x Taking the cube root undoes what cubing does to any real number, and similarly, cubing undoes what taking the cube root does. Example: = 4 and ( 3 4 ) 3 = 4 Similarly, taking the logarithm undoes what using a number as an exponent in an exponential function does (if you use the same base in both cases), and vice versa. Example: 5 b 5 log b (b 5 ) = 5 and 5 log b (5) b log b (5) = 5 Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

10 Comparison: Exponential & Log Fns; Cube and Cube Root Fns Cubing fn vs Cube Root fn: y = 3 x y 3 = x 3 In general, x 3 = x and ( 3 x ) 3 = x. That is, if f (x) = 3 x and g(x) = x 3, then f (g(x)) = x and g(f (x)) = x. In other words, f (x) = 3 x and g(x) = x 3 are inverse functions. Exponential fn vs Logarithm fn: y = log b (x) b y = x In general, log b (b x ) = x and b log b (x) = x. That is, if f (x) = log b (x) and g(x) = b x, then f (g(x)) = x and g(f (x)) = x. In other words, f (x) = log b (x) and g(x) = b x are also inverse functions. Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

11 Inverse Functions Notation: If f and g are inverses of each other, we write g(x) = f 1 (x). Examples of Inverse Functions f (x) = x 3 f 1 (x) = 3 x f f 1 (x) = f ( ) ( 3 (x) = 3 ) 3 (x) = x and f 1 f (x) = f 1 (x 3 ) = 3 x 3 = x f (x) = 2x + 5, f 1 (x) = x 5 2 ( ) ( ) x 5 x 5 f f 1 (x) = f = and f 1 f (x) = f 1( 2x + 5 ) (2x + 5) 5 = 2 f (x) = 5 x, f 1 (x) = log 5 (x) By definition of the logarithmic function, = 2x 2 = x f f 1 (x) = f ( log 5 (x) ) = 5 log 5 (x) = x and f 1 f (x) = f 1( 5 x) = log 5 (5 x ) = x Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

12 Graphs of Inverse Functions: If a point (a, b) is on the graph of f (x), then that means that f (a) = b. In turn, that must mean that f 1 (b) = f 1 (f (a)) = a, so the point (b, a) is on the graph of f 1 (x). This ends up meaning that the graph of f 1 is the same as the graph of f reflected across the line y = x. Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12

13 Graphs of ln(x), log 3 (x), and log 4 (x), along with those of e x, 3 x, and 4 x Math 101-Calculus 1 (Sklensky) In-Class Work January 29, / 12