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Find: (a) $f g(x)$ ) and; (b) $g(f(x))$.$$f(x)=3 x-2, \quad g(x)=1 / x.$$

(a) $\frac{3-2 x}{x}$(b) $\frac{1}{3 x-2}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Oregon State University

Baylor 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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So this problem starts with just defining some functions with FX s three X minus two and G of X is defined as one over X. So in the first part, when they ask you to find F g of X, what they're really telling you to do is to start with the innermost function. That would be what's inside the parentheses. So we're starting with one over X because that's what G of X is equal to. And then you're going to do f of that function. So when I think of this problem, then what that tells me is toe look at the F function, which is three times X minus two. But what I'm doing, I'm replacing the X and the problem with one over X. So this X gets replaced with one over X. Now, I would leave the answer like this, but some people would actually rewrite the problem, Uh, in several different ways. Maybe you just write as three over X minus two, or you would get the same denominator, which would be three minus two X all over ex Any of these air equal. So then, in part two, part B, when they ask you to find G of ethics. What they're telling you to do is start with the function under this in red, Which is that three X minus two. And then we're gonna go to the G function. And the GI function is one over X. But what we do is replace the X with three experts to so this ex things to be replaced with three X minus two. Um, there's really no reason to simplify that at all. That's perfect.

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