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## Inverse Variation | Modeling and Identifying Inverse Variation

Inverse Variation can be modeled after any equation in the form $xy = k$ where $k \neq 0$. The constant of variation is k.

Figure 1

The table to the left models the time it would take to roller blade 36 miles at various speeds. As the rate increases the amount of time to travel 36 miles decreases. If the rate data is examined more closely and focus is placed on the times of 3, 6 and 12 hours, it should be come clear as the rate is doubled, the amount of time is cut in half.

Since the product is 36 for each pair of numbers, this is an inverse variation problem and the product is the constant of variation for this data. Letting x be the rate and y be the time, one equation to model this data is $xy = 36$. A more useful equation is $y = \frac{36}{x}$. Figure 2 is the graph of the inverse variation modeled in this problem.

Figure 2

#### Critical Thinking – Inverse Variation

Why is $y = \frac{36}{x}$ a more useful equation than $xy = 36$?

### Modeling Inverse Variation

This problem involves finding the equation or function that models a given inverse variation. This is a two step process:

1. Multiply the given x- and y-values
2. Substitute the product into $y = \frac{k}{x}$

Before just memorizing this two step process, it would be better to understand what these two steps are ‘doing’ mathematically. In step 1, multiplying the given x- and y-values calculates the constant of variation. The second step is to substitute the constant of variation into one of the acceptable forms of the inverse variation equation.

#### Example 1 – Modeling Inverse Variation

Suppose that x and y vary inversely and x = 7 and y = 5. Write an equation that models the inverse variation.

$xy = k$   Definition of Inverse Variation

$(7)(5) = k$ Substitution Property of Equality – Step 1 from above

$35 = k$ Simplify

$k = 35$   Symmetric Property of Equality

$y = \frac{k}{x}$   Step 2 from above

$y = \frac{35}{x}$   Final Answer

### Identifying Inverse Variation

When given a table of values, it is often important to be able to identify the equation modeling the data. To determine if a table of values is indeed an inverse variation, all of the data pairs must have the same product. This is because of the definition of an inverse variation and how the constant of variation is calculated. So, to identify a table of values as an inverse variation, one must test the products of all the available data. If the products are all equal, then the data models an inverse variation and by find the products, the constant of variation is already known.

#### Example 2 – Identifying Inverse Variation

Is the relationship between the variables in figure an inverse variation? If so, then write the function that models the inverse variation.

Figure 3

As state above, first find the product of all pairs of numbers:

$2(0.7) = 1.4$

$(0.35) = 1.4$

$14(0.1) = 1.4$.

Since all pairs of corresponding numbers form the same product, this table of values models an inverse variation with the constant of variation being 1.4. Therefore the function to models this data:

$y = \frac{1.4}{x}$.

If I don’t write this at the end of every blog article, I should start. Use the comments section for any questions you may have.

## How to Solve Logarithmic Equations

In this video there are two problems modeled. The first example works with common logs and the second example models an equation with a natural log. The keys to solving any type of logarithmic equation: being able to write the given equation in exponential form or being able to take the log of each side.

To be able to write any logarithmic function in exponential form knowing the definition of logarithm and natural logarithm is key. So here they are…

Definition: Logarithm – The log to the base b of a positive number y i s defined:

If $y = b^x$, then $log_b \hspace{0.1 cm}y = x$.

Definition: Natural Logarithm – If $y = e^x$ , then $log_e \hspace{0.1 cm}y = x$ which is also written as $ln \hspace{0.1 cm}y = x$. The natural logarithmic function is the inverse, written as $y = ln \hspace{0.1 cm}x$.

In other words, if $y = e^x$, then $y = ln \hspace{0.1 cm}x$.

Now that you have reviewed the definition of log and natural log, read through these two examples, then watch the embedded algebra 2 video.

### Example 1 Solving a Logarithmic Equation

This a problem that I assigned to my Algebra 2 class for a review before a quiz on solving logarithmic equations:

$7^{x-3} = 25$ Given

$log 7^{x-3} = log 25$ Take the log of each side

$(x-3)log 7 = log 25$ Product Property

$x-3 = \frac{log25}{log7}$ Division Property

$x = \frac{log25}{log7}+3$ Addition Property

$x \approx 4.6542$ Use a calculator

As you can see in this example, you can easily solve an equation with multiple log in it. All you need to do is use the properties of logarithms and that taking the log of each side is a legal mathematical move. The properties for common logs are used with natural logs.

### Example 2 Solving a Natural Logarithmic Equation

Solve the following equation.

$2 \cdot ln \hspace{0.1 cm}x + 3 \cdot ln \hspace{0.1 cm} 2 = 5$ Given

$ln \hspace{0.1 cm}x^2 + ln \hspace{0.1 cm}2^3 = 5$ Power Property

$ln \hspace{0.1 cm}x^2 + ln \hspace{0.1 cm}8 = 5$ Simplify

$ln \hspace{0.1 cm}8x^2 = 5$ Product Property

$8x^2 = e^5$ Write in Exponential Form

$x^2 = \frac{e^5}{8}$ Division Property

$x = \sqrt{ \frac{e^5}{8}}$ Square Root Property

$x \approx 4.3072$ Use a calculator

Solving logarithmic equations with natural log is easy if you can use the properties of natural logs and you know how to write a natural logarithmic equation in exponential form.

## Findng Inverses of Formulas and Compositions of Inverse Functions

In this video math lesson, working with inverse functions is discussed. It is good to remember, when finding the the inverse of a formula, DO NOT SWITCH THE VARIABLES. The first example models finding the inverse of an existing formula:
$d = 16t^2$
$16t^2 = d$ Symmetric Property of Equality
$t^2 = \frac{d}{16}$ Division Property of Equality
$\sqrt {t^2} = \sqrt{ \frac{d}{16}}$ Inverse of Square is Square Root
$t^ = \frac{ \sqrt{d}}{4}$ Simplify Fraction to Simplest Radical Form

I know some readers may not be able to follow the above problem. The same problem, but with the example in the video, the auditory learner can benefit too.

The second example models finding the composition of a function and its inverse. This can be written as $f^{-1}(f(x))$ or $f(f^{-1}(x))$. In both cases are equal to the value of x. Performing the composition of a function and its inverse gives the value you started with. You will see in the video, how simple this process is.

## Find the Domain of a Function and Its Inverse

In this post, I have embedded an Algebra 2 Video Math Lesson about the function $f(x) = \sqrt {2x+2}$. Before considering the function, the video defines $f^{-1}$ as the inverse of f or as f inverse.

The video models how to:

1. Find the domain and range of $f(x) = \sqrt {2x+2}$
2. Find $f^{-1}$
3. Find the domain and range of $f^{-1}$
4. Determine if $f^{-1}$ is a function

If you have any questions regarding the video lesson or other questions regarding finding the inverse of a function, use the comments section.

## How To Find and Graph the Inverse of a Function

In this video Algebra 2 Math Lesson, I model how to graph a quadratic function and its inverse. The process starts with graphing a parabola in the form $y=a^2+c$. The vertex is given by (0,c). After graphing the parabola with four additional points, I create the inverse’s graph by moving the points about y = x line. Finally, I find the inverse of the original function.

If this video math lesson on finding the inverse of a function and then graphing it helped you, leave a comment. Heck, if it didn’t help leave a comment and let me know how to make it better.

## Algebra 2 Chapter 7 Practice Test Answer Key

This post is primarily for my Algebra 2 students. It is the Answer Key to the Chapter 7 test on Radical Functions and Rational Exponents.

Topics of the test include: roots and radical expressions, multiplying and dividing radical expressions, binomial radical expressions, rational exponents, solving square root and other radical equations, function operations and inverse relations and functions.

The file below contains the work I would show to solve the problems and it is in pdf format. If you have any questions, use the comment section to post them.

Algebra 2 Chapter 7 Practice Test Answer Key

Regards,

Mr. Pi

## How to Find the Inverse of a Relation and an Equation

This video lesson covers the definition of the inverse of a relation and reviews the concept of how to find the inverse of a relation.

Example 1 covers how to find the inverse of a relation from a table of values and provides a good visual of a relation that is a function and its inverse is not a function.

Before example 2, a discussion about how switching the x and the y in the equation is the best method for finding the inverse of a function. Example 2 models how to use the x and y switcheroo to find the inverse of a given function.