Question

warm-up (7 minutes)find the key features of the function f(x) here1. domain2. range3. x-intercept4. y-intercept5. minimum6. maximum7. increasing8. decreasing9. f(1)=figure 10: cube root function $f(x) = sqrt3{x}$

Explanation:

Step1: Identify domain of $\sqrt[3]{x}$

Cube roots accept all real numbers, so domain is all real $x$.

Step2: Identify range of $\sqrt[3]{x}$

Cube roots output all real numbers, so range is all real $y$.

Step3: Find x-intercept

Set $f(x)=0$: $\sqrt[3]{x}=0 \implies x=0$.

Step4: Find y-intercept

Set $x=0$: $f(0)=\sqrt[3]{0}=0$.

Step5: Check for minimum value

The function extends to $-\infty$ as $x\to-\infty$, so no minimum.

Step6: Check for maximum value

The function extends to $+\infty$ as $x\to+\infty$, so no maximum.

Step7: Find increasing interval

The cube root function has a positive derivative $\frac{1}{3x^{2/3}}$ for all $x
eq0$, so it increases everywhere.

Step8: Find decreasing interval

The function never has a negative slope, so no decreasing interval.

Step9: Calculate $f(1)$

Substitute $x=1$: $f(1)=\sqrt[3]{1}=1$.

Answer:

1. Domain: All real numbers, or $(-\infty, \infty)$
2. Range: All real numbers, or $(-\infty, \infty)$
3. X-intercept: $(0, 0)$
4. Y-intercept: $(0, 0)$
5. Minimum: None
6. Maximum: None
7. Increasing: $(-\infty, \infty)$
8. Decreasing: None
9. $F(1)=1$