Solved Problems: Functions from Verbal Descriptions Grade 8

📚 Understanding Functions from Verbal Descriptions

In mathematics, a function is a relationship between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Often, these relationships are described verbally, and our task is to translate these descriptions into mathematical expressions.

📜 Historical Context

The concept of a function has evolved over centuries. Early notions involved geometric relationships, but the formal definition emerged with the development of calculus in the 17th century. Mathematicians like Leibniz and Bernoulli contributed to standardizing the notation and understanding of functions as we know them today.

🔑 Key Principles

• 🔍 Identify Variables: Determine the independent variable (input) and the dependent variable (output). For example, if a problem discusses the cost of items based on quantity, quantity is the independent variable (often $x$), and cost is the dependent variable (often $y$).
• 🔗 Recognize Relationships: Look for keywords that indicate mathematical operations. "Per" suggests division, "sum" suggests addition, "product" suggests multiplication, and "difference" suggests subtraction.
• 📝 Write the Equation: Express the relationship between the variables using mathematical symbols. If $y$ is dependent on $x$, the equation will often be in the form $y = f(x)$.
• 💡 Define the Domain: Consider the possible values of the independent variable. For example, if $x$ represents the number of items, it cannot be negative.

🌍 Real-World Examples

Example 1: Taxi Fare

A taxi charges a flat fee of $3 plus $2 per mile. Write a function that represents the total cost of a taxi ride.

1. Variables:
   • $x$ = number of miles
   • $y$ = total cost
2. Relationship: The total cost is the sum of the flat fee and the cost per mile.
3. Equation: $y = 2x + 3$

Example 2: Earning Money

You earn $15 per hour. Write a function that represents your total earnings.

1. Variables:
   • $x$ = number of hours worked
   • $y$ = total earnings
2. Relationship: Total earnings is the product of the hourly rate and the number of hours worked.
3. Equation: $y = 15x$

Example 3: Area of a Rectangle

The length of a rectangle is 5 more than its width. Express the area of the rectangle as a function of its width.

1. Variables:
   • $w$ = width of the rectangle
   • $l$ = length of the rectangle = $w + 5$
   • $A$ = area of the rectangle
2. Relationship: Area of a rectangle is the product of its length and width.
3. Equation: $A = w(w + 5) = w^2 + 5w$

✍️ Practice Quiz

Translate the following verbal descriptions into functions:

1. A store sells apples for $2 each. Write a function for the total cost of buying apples.
2. A phone company charges a monthly fee of $20 plus $0.10 per minute of call time. Write a function for the total monthly bill.
3. The perimeter of a square is four times the length of one side. Write a function that represents the perimeter of a square.

✅ Conclusion

Translating verbal descriptions into functions is a fundamental skill in mathematics. By carefully identifying variables, recognizing relationships, and writing equations, you can model real-world situations mathematically. Practice is key to mastering this skill.