Composite functions
Let ƒ(x) = x³, g (x) = sin x an...

Question

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .

Find h (ƒ (x)).

Accepted Answer

Welcome back, everyone. In this problem, we want to simplify the composite function g of f of x given that f of x equals x minus 3, all raised to the 5th power and the g of x equals the square root of x. For our answer choices, a is the square root of the expression x minus 3. B is the square root of x minus 3 all raised to the 5th power. C is x minus 3 all squared and the d is the product of x minus 3 all squared and the the square root of the expression x minus 3. Now, we want to simplify the composite function, g of f of x. And recall that for a composite function, in this case, g of f of x, it means that we're putting the function f of x into the function g of x. In other words, we're evaluating the innermost function first. Note that g of x equals the square root of x. So what we're going to do is anywhere we see our input value x, we're going to put f of x. So then, that tells us that g of f of x so let me use a different color here. G of f of x is now going to be equal to the square root of not x but whatever f of x is and f of x is x minus 3 all raised to the 5th power. Now, we can simplify this some more because we're finding the square root of something raised to the 5th power. So we could rewrite our expression as the square root of x minus 3 to the 4th power multiplied by x minus 3. Now in that sense, we can also rewrite it as the square root of x minus 3 raised to the 4th power multiplied by the square root of x minus 3. And we can do that because as you can recall, if we're finding the square root of the product, it's the same thing as multiplying the square root of one term by the square root of the other term. So we apply that here to rewrite our product or the square root of our product. Now, we can go ahead and simplify even more because the square root of something raised to the 4th power is the same thing as writing that expression to the second power And we're still multiplying by the square root of x minus 3. So in its simplest form, g of f of x equals the product of x minus 3 all squared and the square root of the expression x minus 3. When we look back in our answer choices, that would have been answer choice d. Thanks a lot for watching everyone. I hope this video helped.

Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .
Find h (ƒ (x)).

Key Concepts

Composite Functions

Function Notation

Basic Functions and Their Properties

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