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Chapter 1: Problem 36

Sketch the graph of the function. $$f(x)=\left\\{\begin{array}{ll}\sqrt{4+x}, & x<0 \\\\\sqrt{4-x}, & x \geq 0\end{array}\right.$$

Short Answer

Expert verified
The graph of the given piecewise function will start from the point (-4,0), increase until the point (0,2), and then decrease until the point (4,0).

Step by step solution

01

Define the Domains

The function is divided into two sub-functions each defined on different domains. For \(x < 0\), \(f(x) = \sqrt{4 + x}\) and for \(x \geq 0\), \(f(x) = \sqrt{4 - x}\).
02

Graphing the First Function

Start by graphing the function \(y = \sqrt{4 + x}\) for the domain \(x < 0\). As the square root of a negative number is not real, only consider values where the term inside the square root is non-negative, i.e., \(4 + x \geq 0\), which implies \(x \geq -4\). So, for the first function, the relevant domain is \(-4 \leq x < 0\). The graph of \(y = \sqrt{4 + x}\) will start from the point \(-4, 0\) and increase as \(x\) approaches \(0\) from the left.
03

Graphing the Second Function

Next, graph the function \(y = \sqrt{4 - x}\) for the domain \(x \geq 0\). Only consider values where the term inside the square root is non-negative, i.e., \(4 - x \geq 0\), which implies \(x \leq 4\). So, for the second function, the relevant domain is \(0 \leq x \leq 4\). The graph of \(y = \sqrt{4 - x}\) will start from the point \(0, 2\) (since \(\sqrt{4} = 2\)) and decrease as \(x\) approaches \(4\) from the left.
04

Combine the Graphs

Now combine these two graphs to form the graph of the given piecewise function on their respective domains.

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