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Mathematical Models: Building and Analyzing Functions

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Section 1.6 Mathematical Models: Building Functions

Introduction to Building and Analyzing Functions

Mathematical modeling is a crucial skill in precalculus, allowing us to translate real-world scenarios into mathematical language using functions. This process involves identifying independent and dependent variables, constructing a function to describe their relationship, and analyzing the resulting model for specific values and domains.

Building Functions from Verbal Descriptions

Translating Real-World Problems into Functions

  • Independent Variable: The variable that represents the input or cause (often denoted as x).

  • Dependent Variable: The variable that represents the output or effect (often denoted as y or another letter).

  • Function Rule: The equation or formula that relates the independent and dependent variables.

To build a function, assign symbols to the variables and use the information given to write an equation that models the situation.

Examples of Building and Analyzing Functions

Example 1: Distance from the Origin to a Point on a Graph

Consider a point P = (x, y) on the graph of a function. We are interested in expressing the distance d from P to the origin O = (0, 0) as a function of x.

  • Distance Formula: The distance from P = (x, y) to the origin is given by:

  • If y is given as a function of x (e.g., y = x^2 - 3), substitute to express d in terms of x:

  • Evaluating the Function:

    • If x = 0:

    • If x = 2:

Graph of y = x^2 - 3 with point P(x, y) and distance d to the origin

Example 2: Area of a Rectangle under a Parabola

Suppose a rectangle has one corner at the origin, one on the positive x-axis, one on the positive y-axis, and one in quadrant I on the graph of y = 16 - x^2. We want to express the area A of the rectangle as a function of x and determine its domain.

  • Area Formula: The area of the rectangle is A = x \cdot y.

  • Since the upper-right corner lies on the parabola, y = 16 - x^2:

  • Domain of A(x):

    • Since the rectangle must be in quadrant I, both x > 0 and y > 0.

    • From y = 16 - x^2 > 0, we get x^2 < 16, so -4 < x < 4.

    • Combining with x > 0, the domain is 0 < x < 4.

Rectangle under the parabola y = 16 - x^2 with one vertex at the origin

Summary Table: Key Steps in Building Functions

Step

Description

Example

1. Identify Variables

Assign symbols to independent and dependent variables

x = input, y = output

2. Write the Function Rule

Express the dependent variable in terms of the independent variable

y = x^2 - 3

3. Substitute as Needed

Insert the function rule into other formulas

d(x) = \sqrt{x^2 + (x^2 - 3)^2}

4. Determine Domain

Find all valid input values for the model

0 < x < 4

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