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Find two functions $f$ and $g,$ whose composition $f(x)$ ) will result in the given function $h(x)$.$$h(x)=\left(\frac{2 x+3}{2 x-4}\right)^{3}.$$

$$f(x)=x^{3}, g(x)=\frac{2 x+3}{2 x-4}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Missouri State University

Campbell 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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eso We're doing the decomposition function which is basically saying, Hey, we want to split up h of X in to be two separate functions a function inside of a function eso as I look at the original problem, we have two X plus three over two X minus four And that function Cube. What I like to do is just look at where is your inner function? Okay, so I looked for parentheses because over here we have an inner function. So that's kind of my clue that what's inside parentheses is going to be what g of X is to find us. So we're looking at two X plus three over two X minus four. So from here, what I do is I look for the outer function. Well, the outer function is the inter function being cute. But we already took care of the inter function. So sometimes what I do is I tell my son is just like we've already talked about this. So we're done talking with the inter function and I put a giant X there, so that tells my students Okay, well, that means that the outer function, which is F in this problems. The F is the outer peace is X huge because we replace the and are within X So what you could do? You don't have to because this is your final answer is you could do f g of X and see if the equals h of X up here. And it should, because you take the inter function and replace it into the outer function on do you would get the same answer.

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