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Find: (a) $f g(x)$ ) and; (b) $g(f(x))$.$$f(x)=\frac{1}{1-x}, \quad g(x)=\frac{x-1}{x}.$$

(a) $x$(b) $x$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Missouri State University

Oregon State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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01:11

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03:11

So we're doing these composition functions. And the very first one is for FX is 1/1 minus X, and the second function is X minus 1/1, uh, X minus one over X. That's GLX. So, what the composition function is it's an operation where we put one function inside of the other. I've seen teachers explain this. That you go well, right Toe left. I really like that explanation. Um, because I like to think of just the order of operations that you do G of X first. Well, G of X, defined as X minus one over X. Um, so everybody's familiar with grouping symbols. You do the group first. And then what you do is you do the function of f of that which is telling you to take this value for X minus one over X and replace it in for this X. Um So what we have is 1/1 minus X minus one over X. Now, what you can actually do is you can simplify this because you can divide each piece by X eso. What I'm looking at is 1/1 minus x by by X is one And then I'm also going to distribute this minus in here. So it turns into plus one over X on what you should notice is one minus one actually cancels each other out. And in order to divide by a fraction, multiply by the reciprocal, you know, times X over one. So these pieces actually canceled. So the answer to part A is actually equal to X on DWhite. This is a clue for is if the composition function cancels everything else out, you're left with X is thes air actually in versus with each other. Um, if I did the other function part B g of f of X or the quick answer is, Well, if there in verses excuse me, then, uh, it doesn't matter if you do F g of x g of f of X, you get the same answer you get, X. So that's the second answer. But let me show you why you do ffx first, which is that 1/1 minus X. And then you do g of that function, which you replace both exes With that. So 1/1 minus x uh, minus one. All over. 1/1 minus X and just like before in order to divide by fractions, multiply by the reciprocal the reciprocal of one minus that 1/1 minus six. I'm saying something wrong, and then when you distribute that in, the one minus X is canceled there. So you're left with one minus one, and then you have to distribute that negative as well. So it becomes plus X, and lo and behold, the one minus one cancels. You know, after Dex, uh, I think I've said everything I needed you. Okay, that's your answer.

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