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Chapter 1: Problem 73

Find a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=x^{2}+4, g(x)=\sqrt{1-x}$$

Short Answer

Expert verified
\((f \circ g)(x) = 5 - x\), the domain of \(f \circ g\) is \(-\infty, 1]\).

Step by step solution

01

Compute the composition \(f \circ g(x)\)

To find the composite function \((f \circ g)(x)\), replace every instance of 'x' in the function \(f\) with the function \(g(x)\). In this case, \(f(g(x)) = (\sqrt{1-x})^{2} + 4 = 1 - x + 4 = 5 - x.
02

Determine the domain of \((f \circ g)(x)\)

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The square root function is undefined for negative numbers, hence for function \(g(x) = \sqrt{1-x}\), it will only be defined when \(1-x\) is greater than or equal to zero. Solving for \(x\), we obtain \(x \leq 1\). Thus, the domain of \(f \circ g\) is \(-\infty, 1]\).
03

Summary of solutions

The composite function \((f \circ g)(x)\) is \(5 - x\) and the domain of the composite function is \(-\infty, 1]\). This includes all values less than or equal to 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Composition
Understanding function composition is like learning how to combine two recipes into one delightful dish. When we talk about composing two functions, we're essentially creating a new function by applying one function to the result of another.

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