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Problem 5 Medium Difficulty

(a) Write an equation that defines the exponential function with base $ b > 0 $.
(b) What is the domain of this function?
(c) If $ b \neq 1$, what is the range of this function?
(d) Sketch the general shape of the graph of the exponential function for each of the following cases.
(i) $ b > 0 $
(ii) $ b = 1 $
(iii) $ 0 < b < 1 $


a) $f(x)=b^{x}, \quad b>0$
b) $(-\infty, \infty)$
c) $(0, \infty)$
d) See Figures $4(c), 4(b),$ and $4(a),$ respectively.

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Video Transcript

all right. This problem is all about the exponential function and the basic information we need to know, starting with its equation. So we have something of the form. Why equals B to the X where B is greater than zero. Now, let's talk about the domain and range. So let's taken example. Suppose we have y equals two to the X. What numbers? What real numbers could XB So that you get a real output. Could you have to to a negative power, like negative 10 negative 100. Negative 992. Well, yes, you could. Those would all be just small numbers, such as one over to to the 10th. One over to to the 100 one over to to the 9 92nd etcetera. But those are small numbers, but they are real numbers. Could you have to? To a positive power to to the 10th. To to the 100 to to the 99,000. Whatever. Yes, you could. Any positive And could you have 2 to 0? Yes, you could. That would just be one. So any real number can be an exponents on a positive base, like two or any other positive base. So that means that the domain is all real numbers. We can write that as negative infinity to infinity. So how about the range? Well, when you raise your positive base to a power, you're always going to get a positive result. If you are looking back at the example, all of these results are positive as well as if you had to to the 10th. That would be positive two to the 1/100 positive. To do any positive is positive and 2 to 0 was one. So that's positive as well. So that means the range is just the positive values. Why is greater than zero, which we can write in interval notation as zero to infinity? The next thing we want to think about for part D do you know my alphabet D? It comes after c is the sketch of the function. So what will it look like? So what will it look like? If the base is positive and greater than one? What will it look like if the base equals one on what will it look like if the base is between zero and one? So if the base is greater than one. We have exponential growth. If the base is between between zero and one, we have exponential decay. If the base is equal to one, we don't really have exponential growth or decay. We just have something horizontal because it's always going to be at a height of one.

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