Transcript Finding the Inverse of a Linear Function

SECTION
2.2
Modeling with Linear
Functions
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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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Direct Variation, Example 1
Example: w is directly proportional to z. If w = −6
when z = 2,find w when z = −7.
Solution:
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Direct Variation, Example 1
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Direct Variation, Example 2
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:
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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
cont’d
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
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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
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