Calculating the Area of a Rectangle using Functions

Oct 13, 2025 by windailys.com 136 views

Calculating the Area of a Rectangle using Functions

When dealing with functions that represent the dimensions of a rectangle, we can use these functions to calculate the area of the rectangle. The area of a rectangle is given by the product of its length and width. In this case, the length is represented by the function $f(x) = x$ and the width is represented by the function $g(x) = x + 3$ . To find the area, we need to multiply these two functions together.

Multiplying Functions

To multiply two functions, we multiply their corresponding terms. In this case, we have:

$(f \cdot g)(x) = f(x) \cdot g(x) = x \cdot (x + 3)$

Expanding the Product

To expand the product, we need to multiply the terms inside the parentheses by the term outside the parentheses. This gives us:

$(f \cdot g)(x) = x \cdot x + x \cdot 3$

Simplifying the Expression

We can simplify the expression by combining like terms. The term $x \cdot x$ is equal to $x^2$ , and the term $x \cdot 3$ is equal to $3x$ . Therefore, we have:

$(f \cdot g)(x) = x^2 + 3x$

What the Expression Represents

The expression $(f \cdot g)(x) = x^2 + 3x$ represents the area of the rectangle as a function of $x$ . This means that if we plug in a value of $x$ into the expression, we will get the area of the rectangle with a length of $x$ feet and a width of $x + 3$ feet.

Graphing the Expression

To visualize the area of the rectangle as a function of $x$ , we can graph the expression $(f \cdot g)(x) = x^2 + 3x$ . The graph will be a parabola that opens upward, with the vertex at the point $(0, 0)$ .

Real-World Applications

The concept of multiplying functions to find the area of a rectangle has many real-world applications. For example, in architecture, we can use this concept to calculate the area of a building's floor plan. In engineering, we can use this concept to calculate the area of a bridge's deck. In business, we can use this concept to calculate the area of a company's warehouse.

Conclusion

In conclusion, the expression $(f \cdot g)(x) = x^2 + 3x$ represents the area of a rectangle as a function of $x$ . This expression can be used to calculate the area of a rectangle with a length of $x$ feet and a width of $x + 3$ feet. The concept of multiplying functions to find the area of a rectangle has many real-world applications, and is an important tool in mathematics and science.

Additional Examples

Here are a few additional examples of how to multiply functions to find the area of a rectangle:

If the length is represented by the function $f(x) = 2x$ and the width is represented by the function $g(x) = x + 2$ , then the area of the rectangle is given by the expression $(f \cdot g)(x) = 2x(x + 2) = 2x^2 + 4x$ .
If the length is represented by the function $f(x) = x^2$ and the width is represented by the function $g(x) = x + 1$ , then the area of the rectangle is given by the expression $(f \cdot g)(x) = x^2(x + 1) = x^3 + x^2$ .

Exercises

Here are a few exercises to practice multiplying functions to find the area of a rectangle:

If the length is represented by the function $f(x) = 3x$ and the width is represented by the function $g(x) = x + 1$ , then find the area of the rectangle.
If the length is represented by the function $f(x) = x^2 + 1$ and the width is represented by the function $g(x) = x + 2$ , then find the area of the rectangle.

Solutions

Here are the solutions to the exercises:

If the length is represented by the function $f(x) = 3x$ and the width is represented by the function $g(x) = x + 1$ , then the area of the rectangle is given by the expression $(f \cdot g)(x) = 3x(x + 1) = 3x^2 + 3x$ .
If the length is represented by the function $f(x) = x^2 + 1$ and the width is represented by the function $g(x) = x + 2$ , then the area of the rectangle is given by the expression $(f \cdot g)(x) = (x^2 + 1)(x + 2) = x^3 + 2x^2 + x + 2$ .

Conclusion

In conclusion, the concept of multiplying functions to find the area of a rectangle is an important tool in mathematics and science. By using this concept, we can calculate the area of a rectangle with a length and width represented by functions. The expression $(f \cdot g)(x) = x^2 + 3x$ represents the area of a rectangle as a function of $x$ , and can be used to calculate the area of a rectangle with a length of $x$ feet and a width of $x + 3$ feet.

Calculating the Area of a Rectangle using Functions

Multiplying Functions

To multiply two functions, we multiply their corresponding terms. In this case, we have:

$(f \cdot g)(x) = f(x) \cdot g(x) = x \cdot (x + 3)$

Expanding the Product

To expand the product, we need to multiply the terms inside the parentheses by the term outside the parentheses. This gives us:

$(f \cdot g)(x) = x \cdot x + x \cdot 3$

Simplifying the Expression

We can simplify the expression by combining like terms. The term $x \cdot x$ is equal to $x^2$ , and the term $x \cdot 3$ is equal to $3x$ . Therefore, we have:

$(f \cdot g)(x) = x^2 + 3x$

What the Expression Represents

Q&A

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is given by the product of its length and width.

Q: How do we multiply functions to find the area of a rectangle?

A: To multiply functions to find the area of a rectangle, we multiply the corresponding terms of the two functions together.

Q: What is the expression $(f \cdot g)(x) = x^2 + 3x$ ?

A: The expression $(f \cdot g)(x) = x^2 + 3x$ represents the area of the rectangle as a function of $x$ .

Q: What is the length and width of the rectangle represented by the functions $f(x) = x$ and $g(x) = x + 3$ ?

A: The length of the rectangle is represented by the function $f(x) = x$ , and the width is represented by the function $g(x) = x + 3$ .

Q: How do we simplify the expression $(f \cdot g)(x) = x \cdot (x + 3)$ ?

A: We can simplify the expression by combining like terms. The term $x \cdot x$ is equal to $x^2$ , and the term $x \cdot 3$ is equal to $3x$ . Therefore, we have:

$(f \cdot g)(x) = x^2 + 3x$

Q: What is the real-world application of multiplying functions to find the area of a rectangle?

A: The concept of multiplying functions to find the area of a rectangle has many real-world applications, such as in architecture, engineering, and business.

Q: How do we graph the expression $(f \cdot g)(x) = x^2 + 3x$ ?

A: To graph the expression $(f \cdot g)(x) = x^2 + 3x$ , we can use a graphing calculator or a computer program to visualize the parabola that opens upward, with the vertex at the point $(0, 0)$ .

Q: What is the solution to the exercise where the length is represented by the function $f(x) = 3x$ and the width is represented by the function $g(x) = x + 1$ ?

A: The solution to the exercise is given by the expression $(f \cdot g)(x) = 3x(x + 1) = 3x^2 + 3x$ .

Q: What is the solution to the exercise where the length is represented by the function $f(x) = x^2 + 1$ and the width is represented by the function $g(x) = x + 2$ ?

A: The solution to the exercise is given by the expression $(f \cdot g)(x) = (x^2 + 1)(x + 2) = x^3 + 2x^2 + x + 2$ .