Question

identify the graphical characteristics for the following radical function.
$f(x)=3\sqrt3{-(x - 1)}-3$

where is the $x$-intercept? if there is no $x$-intercept, enter dne.
$x = $

where is the $y$-intercept? if there is no $y$-intercept, enter dne.
$y = $

what is the domain? enter your answer as an interval.
enter your answer

what is the range? enter your answer as an interval.
enter your answer

as the value of $x$ approaches positive infinity, the value of $f(x)$ approaches:
enter your answer

as the value of $x$ approaches positive infinity, the value of $f(x)$ approaches:
enter your answer

as the value of $x$ approaches negative infinity, the value of $f(x)$ approaches:
enter your answer

in what interval is the function increasing? if there is more than one interval, use union notation (,) to list them. if there is no such interval, enter dne.
enter your answer

in what interval is the function decreasing? if there is more than one interval, use union notation (,) to list them. if there is no such interval, enter dne.
enter your answer

Explanation:

Step1: Find x-intercept (f(x)=0)

Set $3\sqrt[3]{-(x-1)} - 3 = 0$
$\sqrt[3]{-(x-1)} = 1$
$-(x-1) = 1^3 = 1$
$-x + 1 = 1$
$x = 0$

Step2: Find y-intercept (x=0)

$f(0) = 3\sqrt[3]{-(0-1)} - 3 = 3\sqrt[3]{1} - 3 = 3(1)-3=0$

Step3: Determine domain

Cube roots accept all real $x$, so domain is $(-\infty, \infty)$

Step4: Determine range

Cube roots output all real numbers, so range is $(-\infty, \infty)$

Step5: Limit as $x\to+\infty$

$\sqrt[3]{-(x-1)} = \sqrt[3]{-x+1} \to -\infty$
$3(-\infty)-3 \to -\infty$

Step6: Limit as $x\to-\infty$

$\sqrt[3]{-(x-1)} = \sqrt[3]{-x+1} \to +\infty$
$3(+\infty)-3 \to +\infty$

Step7: Find increasing/decreasing intervals

Rewrite $f(x) = 3(-(x-1))^{1/3} - 3 = 3(1-x)^{1/3} - 3$
Take derivative: $f'(x) = 3 \cdot \frac{1}{3}(1-x)^{-2/3}(-1) = -\frac{1}{(1-x)^{2/3}}$
$f'(x) < 0$ for all $x
eq 1$, so function is always decreasing.

Answer:

• $x$-intercept: $x = 0$
• $y$-intercept: $y = 0$
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$
• As $x\to+\infty$, $f(x)$ approaches: $-\infty$
• As $x\to-\infty$, $f(x)$ approaches: $+\infty$
• Increasing interval: DNE
• Decreasing interval: $(-\infty, \infty)$