Question

1. let y = √(81 - x²).

a. what is the domain of the function?
b. what is the range of the function?

1. y=(x - 4)(x + 3)(x + 5)

a. what is the domain of the function?
b. what is the range of the function?

1. the battery charge of a cell phone is tracked with a function over the course of one day. graph a possible function below that correctly represents the domain and range of the situation.

Explanation:

4.

Step1: Find domain for $y = \sqrt{81 - x^{2}}$

For square - root function, the expression inside the square - root must be non - negative. So we set $81 - x^{2}\geq0$.
This can be rewritten as $x^{2}-81\leq0$, and factored to $(x + 9)(x - 9)\leq0$. The solutions of the inequality are $-9\leq x\leq9$.

Step2: Find range for $y = \sqrt{81 - x^{2}}$

The maximum value of $81 - x^{2}$ occurs when $x = 0$, and $81 - x^{2}=81$. So $y=\sqrt{81 - x^{2}}\leq9$. Also, since $\sqrt{81 - x^{2}}\geq0$, the range is $0\leq y\leq9$.

Step1: Find domain for $y=(x - 4)(x + 3)(x + 5)$

This is a polynomial function. Polynomial functions are defined for all real numbers. So the domain is all real numbers, $(-\infty,\infty)$.

Step2: Find range for $y=(x - 4)(x + 3)(x + 5)$

Since it is a cubic polynomial with a positive leading coefficient (the leading term when expanded is $x^{3}$), as $x\to-\infty$, $y\to-\infty$ and as $x\to\infty$, $y\to\infty$. So the range is all real numbers, $(-\infty,\infty)$.

The domain represents the time in a day. Time $t$ ranges from $0$ to $24$ hours, so the domain is $[0,24]$. The range represents the battery charge percentage. The battery charge percentage $p$ ranges from $0\%$ to $100\%$, so the range is $[0,100]$. A possible function could start at $100$ (fully charged at the start of the day), decrease over time, and end at some non - negative value (e.g., $20$ if it's not fully drained by the end of the day). The graph would be a curve starting at $(0,100)$ and ending at $(24,k)$ where $0\leq k\leq100$, and decreasing overall.

Answer:

a. Domain: $[-9,9]$
b. Range: $[0,9]$

5.