Exponential functions from tables & graphs
Consider the following table of values for a linear function f of x is equal to mx plus b and an exponential function g of x is equal to a times r to the x. Write the equation for each function. And so they give us, for each x-value, what f of x is and what g of x is. And we need to figure out the equation for each function and type them in over here. So I copy and pasted this problem on my little scratchpad. So let's first think about the linear function. And to figure out the equation of a line or a linear function right over here, you really just need two points. And I always like to use the situation when x equals 0 because that makes it very clear what the y-intercept is going to be. So, for example, we can say that f of 0 is going to be equal to m times 0 plus b. Or this is just going to be equal to b. And they tell us that f of 0 is equal to 5. b is equal to 5. So we immediately know that this b right over here is equal to 5. Now, we just have to figure out the m. We have to figure out the slope of this line. So just as a little bit of a refresher on slope, the slope of this line is going to be our change in y-- or our change in our function I guess we could say, if we say that this y is equal to f of x-- over our change in x. And actually, let me write it that way. We could write this as our change in our function over our change in x if you want to look at it that way. So let's look at this first change in x when x goes from 0 to 1. So we finish at 1. We started at 0. And f of x finishes at 7 and started at 5. So when x is 1, f of x is 7. When x is 0, f of x is 5. And we get a change in our function of 2 when x changes by 1. So our m is equal to 2. And you see that. When x increases by 1, our function increases by 2. So now we know the equation for f of x. f of x is going to be equal to 2 times 2x plus b, or 5. So we figured out what f of x is. Now we need to figure out what g of x is. So g of x is an exponential function. And there's really two things that we need to figure out. We need to figure out what a is, and we need to figure out what r is. And let me just rewrite that. So we know that g of x-- maybe I'll do it down here. g of x is equal to a times r to the x power. And if we know what g of 0 is, that's a pretty useful thing. Because r to the 0th power, regardless of what r is-- or I guess we could assume that r is not equal to 0. People can debate what 0 to the 0 power is. But if r is any non-zero number, we know that if you raise it to the 0 power, you get 1. And so that essentially gives us a. So let's just write that down. g of 0 is a times r to the 0 power, which is just going to be equal to a times 1 or a. And they tell us what g of 0 is. g of 0 is equal to 3. So we know that a is equal to 3. So so far, we know that our g of x can be written as 3 times r to the x power. So now we can just use any one of the other values they gave us to solve for r. For example, they tell us that g of 1 is equal to 2. So let's write that down. g of 1, which would be 3 times r to the first power, or just 3-- let me just write it. It could be 3 times r to the first power, or we could just write that as 3 times r. They tell us that g of 1 is equal to 2. So we get 3 times r is equal to 2. Or we get that r is equal to 2/3. Divide both sides of this equation by 3. So r is 2/3. And we're done. g of x is equal to 3 times 2/3. Actually, let me just write it this way. 3 times 2/3 to the x power. You could write it that way if you want, any which way. So 3 times 2/3 to the x power, and f of x is 2x plus 5. So let's actually just type that in. So f of x is 2x plus 5. And we can verify that that's the expression that we want. And g of x is 3 times 2 over 3 to the x power. And let me just verify that that's what I did there. I have a short memory. All right. Yeah, that looks right. All right. Let's check our answer. And we got it right.