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Modeling with Linear Functions

SECTION 2.2

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Learning Objectives

1 Determine if two quantities are directly proportional

2 Construct linear models of real-world data sets and use them to predict results

3 Find the inverse of a linear function and interpret its meaning in a real-world Context

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Recognizing When to Use a Linear Model

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Recognizing When to Use a Linear Model

Several key phrases alert us to the fact that a linear model may be used to model a data set.

Some of the simplest linear models to construct are those that model direct proportionalities, where one quantity is a constant multiple of another quantity.

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Recognizing When to Use a Linear Model

Solving the equation y = kx for k yields

Thus another way to define direct proportionality is to say that two quantities are directly proportional if the output divided by the input is a constant.

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Example: w is directly proportional to z. If w = −6 when z = 2,find w when z = −7.

Solution:

Direct Variation, Example 1

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Direct Variation, Example 1

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Garth’s front lawn is a rectangle measuring 120 feet by 40 feet. If a 25lb. bag of “Weed and Feed” will treat 2,000 square feet, how many bags must Garth buy to treat his lawn? (Assume that he must buy whole bags.)

Solution:

Direct Variation, Example 2

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Direct Variation, Example 2

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Inverses of Linear Functions

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Inverses of Linear Functions

We have known that the domain of a function f was the range of its inverse function f

–1 and the range of the function f was the domain of its inverse function f

–1.

This notion is represented as follows.

The phrase “y is the function of x” corresponds with the phrase “x is the inverse function of y.”

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Inverses of Linear Functions

Symbolically, y = f (x) is related to x = f –1(y).

Recognizing this relationship between a function and its inverse is critical for a deep understanding of inverse functions, especially in a real-world context.

Many students struggle with the concept of inverse functions.

To help you get a better grasp on this concept, we will work a straightforward example before summarizing the process of finding an inverse of a linear function.

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Example 5 – Finding the Inverse of a Linear Function

Find the inverse of the function .

Solution:In the function we are given, x is the independent variable and y is the dependent variable. We solve this equation for x.

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Example 5 – Solution

In this new equation, y is the independent variable and x is the dependent variable. We can write the original function as

Similarly, we can write the inverse function as

The function is the inverse of

cont’d

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Inverses of Linear Functions

We now summarize the process of finding the inverse of a linear function. Observe that since dividing by 0 is undefined, this process only works for linear functions with nonzero slopes.

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Inverses of Linear Functions

We have known that horizontal lines can be written in the form y = b.

The inverse of a horizontal line is a vertical line x = a; however, a vertical line is not a function. Therefore, only linear functions with nonzero slopes have an inverse function.

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Inverses of Linear Functions Example

Based on data from June 2006, the forecast for maximum 5-day snow runoff volumes for the American River at Folsom, CA, can be modeled by v = f(t) = −2.606t + 131.8 thousand acre-feet,

where t is the number of days since the end of May 2006. That is, the model forecasts the snow runoff for the 5-day period beginning on the selected day of June 2006. Find the inverse function.

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Inverses of Linear Functions Example

v = f(t) = −2.606t + 131.8

Solution:

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Exit Ticket:

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Inverses of Linear Functions Example