Calculus Examples

$f (x) = {\begin{cases} 2 x, x < 1 \\ 5, x = 1 \\ x^{2}, x > 1 \end{cases}$

Step 1

Find the domain.

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Step 1.1

Since the domain of $2 x$ is all real numbers, the domain of this piece of the function is its restriction, $x < 1$ .

$(- \infty, 1)$

Step 1.2

Since the domain of $5$ is all real numbers, the domain of this piece of the function is its restriction, $x = 1$ .

$[1, 1]$

Step 1.3

Since the domain of $x^{2}$ is all real numbers, the domain of this piece of the function is its restriction, $x > 1$ .

$(1, \infty)$

Step 1.4

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

$(- \infty, 1) \cup (1, \infty)$

Step 2

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

$(- \infty, \infty)$

Step 3

List the domain and range of the piecewise.

Domain: $(- \infty, 1) \cup (1, \infty)$

Range: $(- \infty, \infty)$

Step 4