(a) write formulas for f ∘g and g ∘f and (b) find the domain of each. f(x)=x^2, g(x)=1-√(x)

Name: Problem 18, Chapter 1: University Calculus: Early Transcendentals
Uploaded: 2020-06-29T17:51:56-08:00
Duration: 2 min 4 s
Channel: Charles Carter
Description: (a) write formulas for f ∘g and g ∘f and (b) find the domain of each. f(x)=x^2, g(x)=1-√(x)

step 1 Okay, so finding F of G of X means we're going to plug G of X into F. And so we're going to do F

(a) write formulas for f ∘g and g ∘f and (b) find the domain...

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Key Concepts

Function Composition

This concept involves creating a new function by applying one function to the results of another, represented as (f ? g)(x) = f(g(x)). It requires a thorough understanding of how the output of the inner function becomes the input for the outer function, and is foundational in analyzing complex function behaviors.

Composite Function Domain

The domain of a composite function is determined by first considering the domain of the inner function and then ensuring that the outputs of the inner function fall within the domain of the outer function. This layered approach ensures that every step in the composition is valid, meaning all restrictions present in the original functions are taken into account.

Domain Restrictions of Square Root Functions

Since the square root function is only defined for nonnegative inputs in the realm of real numbers, any function involving a square root must restrict its domain accordingly. This concept is critical when determining the allowed inputs for composite functions that include square roots, as it directly impacts the overall domain.

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