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Given a function \{3x, \text{ if } x \leq 0 \mid x^3, \text{ if } x > 0\} use the graph of $f$ to find the domain of $f'$ .

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Find the vertex of the transformed function

Full solution

Q. Given a function

\{3x, \text{ if } x \leq 0 \mid x^3, \text{ if } x > 0\}

use the graph of

f

to find the domain of

f'

.

Identify Function: Identify the given function $f(x)$ : $f(x) = \begin{cases} 3x, & \text{if } x \leq 0 \ x^3, & \text{if } x > 0 \end{cases}$
Determine Domain: Determine the domain of $f(x)$ :
For $x \leq 0$ , $f(x) = 3x$
For x > 0 , $f(x) = x^3$
Find Derivative: Find the derivative $f'(x)$ :
For $x \leq 0$ , $f'(x) = 3$
For x > 0 , $f'(x) = 3x^2$
Domain of Derivative: Determine the domain of $f'(x)$ : For $x \leq 0$ , $f'(x) = 3$ (defined for all $x \leq 0$ ) For x > 0, $f'(x) = 3x^2$ (defined for all x > 0)
Combine Domains: Combine the domains: Domain of $f'$ is all real numbers ( $x$ in $(-\infty, \infty)$ )

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