Real-World Examples of Domain and Range in Pre-Calculus Applications

📚 Quick Study Guide

• 🔢 Domain: The set of all possible input values (x-values) for which a function is defined. Think of it as what you're allowed to plug into your equation.
• 📈 Range: The set of all possible output values (y-values) that a function can produce. This is what you get out of the equation after plugging in the domain.
• 🌱 Real-World Examples: We'll see how these concepts apply to things like projectile motion, population growth, and even the cost of manufacturing items!
• 💡 Important Note: Pay attention to any restrictions on the input values in real-world scenarios, such as time not being negative or a limited amount of resources.
• 📐 Mathematical Notation: Domain and range are often expressed using interval notation or set notation. For example, $[0, \infty)$ means all numbers greater than or equal to 0.

🧪 Practice Quiz

1. What is the domain of a function representing the height of a ball thrown in the air, where $t$ represents time, considering the ball starts at $t=0$ and lands at $t=5$ seconds?
   1. $(-\infty, \infty)$
   2. $(0, 5)$
   3. $[0, 5]$
   4. $[0, \infty)$
2. A company's profit $P(x)$ is modeled by $P(x) = -2x^2 + 100x - 50$, where $x$ is the number of units sold. What is the reasonable domain for $x$ in this context?
   1. $(-\infty, \infty)$
   2. $[0, \infty)$
   3. $[0, 50]$
   4. $(-50, 50)$
3. The population of a bacteria colony is modeled by $N(t) = 100e^{0.2t}$, where $t$ is time in hours. What is the range of $N(t)$ for $t \geq 0$?
   1. $[0, \infty)$
   2. $[100, \infty)$
   3. $(0, \infty)$
   4. $(-\infty, \infty)$
4. A farmer wants to fence a rectangular garden. He has 200 feet of fencing. If $x$ is the length of one side, express the area $A(x)$ as a function of $x$. What is the domain of $A(x)$ in this context?
   1. $(0, 100)$
   2. $[0, 100]$
   3. $(0, \infty)$
   4. $(-\infty, \infty)$
5. The cost $C(x)$ to produce $x$ widgets is given by $C(x) = 5x + 100$. If the company can produce a maximum of 500 widgets, what is the range of $C(x)$?
   1. $[0, \infty)$
   2. $[100, 2600]$
   3. $[0, 500]$
   4. $[5, 500]$
6. A function $f(x)$ represents the temperature (in degrees Celsius) of a chemical reaction $x$ minutes after it starts. If the reaction is observed for one hour, what is the most appropriate domain for $f(x)$?
   1. $[0, 1]$
   2. $[0, 60]$
   3. $[1, 60]$
   4. $(-\infty, \infty)$
7. The height $h$ of a projectile launched at an angle is given by $h(x) = -0.01x^2 + x$, where $x$ is the horizontal distance traveled. What is the domain of $h(x)$ if the projectile lands when $h(x) = 0$?
   1. $[0, 100]$
   2. $[0, \infty)$
   3. $(-\infty, \infty)$
   4. $[0, 50]$