find the domain and range for the cube root function. state the x- and y-intercepts, if they…

Explanation:

Step1: Analyze cube root domain

Unlike square roots, cube roots are defined for all real numbers, since we can take the cube of any positive, negative, or zero real number. The function $f(x)=\sqrt[3]{8x}$ has no restrictions on the input $x$.

Step2: Define domain inequality

All real numbers can be written as the inequality $-\infty < x < \infty$.

Step3: Find x-intercept

Set $f(x)=0$, solve $\sqrt[3]{8x}=0$. Cube both sides: $8x=0$, so $x=0$. The x-intercept is $(0,0)$.

Step4: Find y-intercept

Set $x=0$, calculate $f(0)=\sqrt[3]{8*0}=0$. The y-intercept is $(0,0)$.

Step5: Analyze cube root range

Cube root functions output all real numbers, since for any real number $y$, we can find an $x$ such that $\sqrt[3]{8x}=y$ (solve for $x$: $x=\frac{y^3}{8}$, which is defined for all real $y$).

Answer:

• Domain of $y=f(x)$: $-\infty < x < \infty$
• Range of $y=f(x)$: $-\infty < y < \infty$
• x-intercept: $(0, 0)$
• y-intercept: $(0, 0)$