In this section, we want to do a little bit of work with exponential functions,
and let's look at some problems that involve finding the value of an exponential function.
If f(x) = 3 to the x, and g(x) = 1/2ˣ, let's find g(0). So, I'll do that by simply substituting 0 for x in the function g.
So, I'll have, 1/2⁰, and I know that that's the number 1. So, g(0) turns out to be 1.
Let's find f(-3). I'll do that by going to the function f and substituting for x, -3.
So, I have 3⁻³, that's going to be 1 over 3³; and 3³ = 27;
so, that would be 1/27. So, exponential functions work just like regular functions in, when I want to find a value in a function here,
I just replace x with that value and simplify. Let's look at one more of these problems. I have the f(2) + g(-2).
While f = 3ˣ, so I make this 3²; + g(-2) is going to be 1/2⁻²
So, I have 3² + 1/2⁻²; 3² = 9; 1/2⁻² well, if I 1/2², I get 1/4;
and I take the reciprocal of that, I get 4. So, I end up with 9 + 4, and 9 + 4 = 13. So, that is f(2) + g(-2);
and I simply substitute for my variable x in the expressions for f and g and simplify what I get.
Next, let's look at the graph of an exponential function. We have y = 2 to the x for our exponential function.
I want to find the graph of this. Let's first go through and make a table of values.
When x = -2, I have y = 2⁻²; 2⁻² is going to be 1 over 2², which will be 1/4.
When I put x = -1 in, I have 2⁻¹, which is 1/2. When I substitute in 0 for x, I have 2⁰, which is 1.
When I substitute in 1, I have 2¹, which is 2. And when I substitute in 2, I have 2² which is 4.
So, let's graph the ordered pair (-2, 1/4); so, I'll go back 2 and up 1/4; x = -1, y = 1/2 so, (-1, 1 1/2);
x = 0, y = 1, so (0,1); x = 1, y = 2, so, (1,2);
and x = 2, y = 4; (2,4). And you can see that if I was to go further here if I said, let x = 3, y would be equal to 8;
x = 4, 2⁴ = 16, so on, and so forth. So, now I'll connect these points with a nice, smooth curve,
and what I get is the graph of the exponential function y = 2ˣ. And you can see, it goes through the point (0,1) on the x-axis
all exponential functions will go through that point, I'm sorry, on the y-axis. (0,1). Let's look at one more problem.
We want to graph the exponential function y = 3ˣ and its inverse.
So first, let's graph y = 3ˣ. When x = 0, y = 1. When x = 1, y = 1, 2, 3 that's 3¹.
When x = 2, y = 3, 4, 5, 6, 7, 8, 9. When x = -1, y = 1/3. When x = -2, y =1/9.
So, I can sketch the graph of that exponential function, y = 3ˣ, real quickly, just like that.
Now, the equation of its inverse is: x = 3ʸ. To find the equation of the inverse from the function itself, I simply exchange x and y.
Now, I could graph that myself separately, or I can use the fact that the graph of a function and its inverse have symmetry
about the line y = x. So, I'll draw in the line y = x looks like this; of course, this would be much nicer on graph paper.
Now, I'll reflect my graph of y = 3ˣ about the line y = x, and I'll have this point: when x = 1, I mean when y = 1, x = 3;
when y = 2, x = 9; when y = -1, x = 1/3, 1/9, so on, and so forth. All those ordered pairs have their coordinates reversed.
And so, here is the graph of x = 3ʸ, which is the inverse function for this function, this exponential function: y = 3ˣ.
So, graphing a function and its inverse is pretty easy if we know the graph of the function.
We simply take that graph, and we reflect it about the line y = x to get the graph of the inverse.
Finding the equation of the inverse from the equation of the function is also easy; we simply exchange x and y.