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Let $f(x)=a x+b$ and $g(x)=c x+d$ determine the conditions on $a, b, c$ and $d$ if $f(g(x))=g(f(x))$.

$$f(d)=g(b)$$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Campbell University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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for this problem, we have been given to generic equations f of X equals X plus B and G X equals C X plus D and in this case, exes air variable. So A, B, C and D just represents, um Constance. And the question we want to answer is what conditions on those constants A. B C D. What did they have to be in order for our two composite functions? F of G of x, n g of f of X? What do those constants have to be to make those two composite functions equal with each other? Well, first of all, let's review what a composite function is normally. Um, is up to this point, we often have a function with a number say f of three. So that means I would go to the F function in place of X. I would put a three well, for a composite function. Instead of having a number in there, we actually have another function, so G f X is the input into my F function. If G was on the outside, that means it Ffx is the input into the GI function. So let's take a look at what we have here. F of G of X. Okay for that piece, that means I'm going to go to the F function, which is a X Plus B, and I'm going to input in the GI function. So C X plus D is what I'm going to substitute in for my ex. And if I want to get rid of those parentheses, that means I have a C X plus. So the first plus a D plus B, that's what the first half of my composite function. Let's try the second one g of f of X. Well, the G's my outside functions. I'm going to go to my GI function, which is C of X plus t. And to substitute in for the X, I'm gonna put my F function, which is a X Plus B. And if I multiply this out, I get a c X plus B C plus de. So those are my two composite functions. Now we're going to set them equal to each other, and I'm going to get a C X plus a d plus B, and that's gonna equal a C X plus B C plus D. Well, I have an a C X on both sides, so I can cancel that. So what I have is a D plus B equaling B C plus D. Well, let's take a look at the first piece a D plus B that looks very similar to my F function. In fact, it's exactly my f function if I'm evaluating it at the point d so f of D is that left hand side? Now let's look at the right hand side B C plus D. Well, that looks exactly like my G of X function. If I allow be to be X Okay, So if I substituted be for X, I would get that right hand side. So these two composite functions air equal as long as this holds true as long as f of D equals G of B.

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