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 the domain of ƒ o g.

Accepted Answer

Welcome back everyone. Given the functions a of x equals negative cosine of x and b of x equals x to the 5th power, we want to determine the domain of the composition of the functions a and b. For our answer choices, a is open parenthesis 0 infinity closed parenthesis. B is open bracket 0 infinity closed parenthesis. C is open parenthesis negative infinity infinity closed parenthesis. And, d is open parenthesis negative infinity zero closed parenthesis. Now, if we're going to figure out the domain of the composite function, or the composition of a and b, recall that that composition means that we're putting b into a. In other words, a of b means we're putting b of x into the function a of x. To simplify the composite function, it means that we must evaluate the innermost function first. So given b of x being equal to x to the 5th power, then for a of x, anywhere we see x, we're going to put that value for b of x. So this tells us then. A of b of x would be equal to the negative cosine of not x but whatever b of x is, which is x to the 5th power. So that would be our composite function. Now, here's the thing. Remember, we want to find the domain of our composite function and the domain really means the input values. Notice that our input value is b of x, which is x to the 5th power. So that means a of b of x would be the same thing as a of x to the 5th power. Now, x can be any real number. And if we raise any real number to the 5th power, then our result still can be any real number. The simplified composite function has no domain constraints or undefined points. Therefore, since the domain is the set of all real numbers, our domain would be between negative infinity and positive infinity. When we look back on our answer choices, that would have been answer choice c. 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 the domain of ƒ o g.

Key Concepts

Composite Functions

Domain of a Function

Trigonometric Functions and Their Domains

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