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

$$f(x)=\frac{4}{\sqrt{x^{2}+4}} g(x)=x$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

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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.

01:46

Find two functions $f$ and…

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There's actually a lot of correct answers in this problem on What we're doing is we're decomposing this function, and I'll go ahead and give you a couple options for your correct answer. Um, and what I tell my students is when you're decomposing, what you're looking for is an inner function. So look for parentheses or some sort of grouping symbol. So how I would do this problem, this is just a mean is I would go ahead and put a group inside the denominator, and, uh and so I'm talking about parentheses. That would be my inner function. So I would call G of X, just the X squared plus four and then to help you out. I hope this helps out. And then ffx would be everything else. So what we're really doing is we're taking out the inter function, and I like Thio. Just put a giant X there to show that Okay, we're going to replace it. The inter function with X, and if you do, if you do have g of X, this works. But I do want to point out another option. So that's one option. Option one is we can have a second option where maybe you group all of the denominator for G of X. Well, if you did that, then everything would be the square root of X squared plus four. But then h of X Would Onley be defined as four over X? Um, I'm sure you could think mawr examples. Um, but here's two options for you and they both work. If you if you were to do f g of X of both of these, you would get the same answers H of x. So there you go.

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