Let f(x)=a x+b and g(x)=b x+a, where a and b are integers. If f(1)=8 and f(g(20))-g(f(20))=-14,

Let f(x)=a x+b and g(x)=b x+a, where a and b are integers....

Key Concepts

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying expressions or equations using arithmetic and algebraic properties. This is essential when working with equations derived from function evaluations and function compositions, as it allows one to combine like terms, substitute values, and ultimately solve for unknown coefficients.

Systems of Equations

Systems of equations are sets of multiple equations that share the same variables. They are solved simultaneously to find the values of these variables. In many algebra problems, including those involving functions, setting up and solving a system of linear equations allows one to determine unknown parameters by using given conditions or constraints.

Linear Functions

Linear functions are functions that can be expressed in the form f(x) = mx + c, where m and c are constants. They have a constant rate of change and their graph is a straight line. Understanding linear functions is crucial for setting up equations based on given function evaluations and for exploring relationships between variables in algebra.

Function Composition

Function composition involves applying one function to the result of another function. It is denoted as (f ? g)(x) = f(g(x)). In many problems involving multiple functions, composing functions and analyzing the resulting expressions is key to deducing relationships between the constants or parameters involved.

Transcript

00:01 Okay, here's another application problem that just is kind of like, you know, the puzzle or thinking exercise based on these two pieces of information that they gave us to start off with.

00:10 F of x is this function.

00:11 G of x is this.

00:13 They also tell us that f of 1 is equal to 8, which implies that a times 1, which is a plus b is equal to 8.

00:23 And then they also tell us that f of g of 20 minus g of 20, equals negative 14.

00:36 All right, cool.

00:38 So we're going to use all these facts to find what they're asking for us.

00:44 You know, after all this, they say, throw out a random question, what's the product of a and b? right.

00:49 So to do that, we need to find a and b and then just multiply those two things together.

00:55 All right.

00:55 So we're going to be using this to kind of help us through a lot as well as kind of composing these two functions.

01:01 So g of 20 is going to be 20b plus a and the f of 20 is going to be 20 a plus b all right one thing i'm going to do here is make sure we can see an error potentially happening here we're going to subtract that whole thing so that might be a whole expression but we're going to subtract that whole thing not just the first term so let's put brackets around that when i plug this in to f i get a times 20b plus a and then add b and then i subtract i'm going to keep these brackets up here for just a little bit when i plug this into g i get b times 20 a plus b and then add a add a this is equal to negative 14 all right all right so what i'm going to do here is distribute this a into here and this this b this b into here again to both terms not just the first one so i have 20 a b plus a squared and then the parentheses are gone because i've distributed plus b i'm going to keep these up for one more step i have 20 a b plus b squared and then plus a this is all equal to negative 14 now i'm going to distribute this negative sign in because that negative has to go to every single term in that expression not just the first one.

02:47 So i'm going to write 20ab plus a squared plus b over here.

02:51 Nothing's changed minus 20ab minus b squared minus a is equal to negative 14.

02:59 All right.

03:02 The cool thing here that i see is i have a negative 20 ab and a positive 20 ab so that takes that out of the equation and i'm going to kind of collect my terms together.

03:12 I have a squared minus b squared plus b minus a is equal to negative 14...

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