Calculus Examples

$f (x) = {\begin{cases} \sqrt{64 - x^{2}}, 0 \leq x < 8 \\ 8, 8 \leq x < 16 \\ 16, x = 16 \end{cases}$

Step 1

Find the domain.

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Step 1.1

Find the domain for $\sqrt{64 - x^{2}}$ .

$[- 8, 8]$

Step 1.2

Compare the domain of $\sqrt{64 - x^{2}}$ with the restriction $0 \leq x < 8$ to see where this piece of the function is defined.

$[0, 8)$

Step 1.3

Since the domain of $8$ is all real numbers, the domain of this piece of the function is its restriction, $8 \leq x < 16$ .

$[8, 16)$

Step 1.4

Since the domain of $16$ is all real numbers, the domain of this piece of the function is its restriction, $x = 16$ .

$[16, 16]$

Step 1.5

Find the domain by taking the union of all of the intervals where the function is defined.

$[0, 16)$

Step 2

The range is the set of all valid $y$ values. Use the graph to find the range.

$(0, 8] \cup [16, 16]$

Step 3

List the domain and range of the piecewise.

Domain: $[0, 16)$

Range: $(0, 8] \cup [16, 16]$

Step 4