About this video
Two functions, f(x) = 3^(-x) - 5 and g(x) = -x^3 - 4x^2 + x + 6, are graphed and intersect at (a, b). Sal explains that this means f(a) = g(a) = b, so x = a is a solution of both 3^(-x) - 5 = b and f(x) = g(x).
You'll be able to
- explain what the intersection point of two graphs means in terms of the functions.
- write equations that the x-coordinate of an intersection solves.
- decide whether a value is a solution to an equation using a graph.
Key words
- point of intersection
- A point where two graphs cross, so it lies on both graphs.
- f(a)
- The output of function f when the input is a.
- solution of an equation
- A value of x that makes the equation true.
Transcript
- [Voiceover] Let f of x equal three to the negative x minus five, and g of x equals negative x to the third minus four x-squared plus x plus six. The graphs of y equals f of x and y equals g of x are shown below, and we see them right over here. This y equals f of x is in that purplish color.
Let me see if I can get that same purplish color. So, that is, in that. So, let me underline it now. So, f of x is in that purplish color. So y equal f of x, that's right over here in purple. And then y equals g of x, that is in blue. So y equals g of x. g of x is defined right over there.
y equals g of x, well that is graphed in the blue. And we see that they intersect at the point a comma b. So there's a couple of ways to think about this. We could say that when x is equal to a, f of x and g of x equal each other. Or, we could say, f of a, and this is coming from this point of intersection.
Let me draw a little arrow here. Or a big arrow. That point of intersection lets us know that f of a, f of a is equal to g of a. Is equal to g of a, which is equal to b. g of a, which is equal to b. They both, if you input a into the function f you're going to get b.
If you input a into the function g, you're going to get b. And so the point a comma b is on both graphs. Both y equals g of x and y equals f of x. And so from here, you could make some interesting statements. For example, you could just say, well, what is f of a?
f of a is 3 to the negative a power minus five, is going to be equal to, what's g of a? g of a is negative a to the third minus four a-squared plus a plus six, so you could say this. And that would be equal to that. Either of those would be equal to b. Alright, I think we've analyzed that a good bit, so now let's actually answer their questions.
And normally I'd suggest you look at the questions before you actually try to solve it, but I just wanted to do this just to really squeeze out as much as we could out of the information they gave us. So they tell us the value x equals a is the solution to which of the following equations?
Select all that apply. So this first one is three to the negative x minus five is equal to b. Well, we already know that, we already know that three to the negative a minus five, Three to the negative a minus is going to be equal to b. Is going to be equal to b.
This over here, this is equivalent to saying, this is equivalent to saying f of x, f of x is equal to b. And we know f of x equals B when x equals a. When x equals a, f of x is equal to b. This expression is equal to b. So we know that that first one is true. Now the second one is just saying, f of x is equal to g of x.
Well, we know that when x is equal to a, f of x is equal to g of x. That f of a is equal to g of a. Because, as a reminder, this right over here is our definition for f of x, and this over here is our definition of g of x. So this is just saying f of x equals g of x, when does that happen?
Well, that happens when x equals a. We already saw it up here. f of a is equal to g of a. They both equal b. And so both of these are going to be equal to each other when x is equal to A. So I will check that one as well.
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