Question

The number of frogs in a certain lake is inversely related to the number of snakes in the lake. If x represents the number of snakes in the lake, then y = 100/x represents the number of frogs in the lake. Describe the reasonable domain and range values. Graph the function.

Answer 1

Both the number of snakes and the number of frogs will be positive, so positive values are reasonable for the domain and range.

Explanation:

Answer 2

1. Domain: The set of natural numbers / Range: The set of natural numbers

2. Shown below

Explanation:

1. Domain and Range.

In this problem, we have that the number of frogs in a certain lake is inversely related to the number of snakes in the lake. Hence we are facing an Inverse Variation, so this means that if the number of snakes in the lake increases then the number of frogs in the lake decreases, because snakes eat frogs! The function that describes this is a rational function defined as:

`y=(100)/(x)`

Where:

`x: \ represents \ the \ number \ of \ snakes \ in \ the \ lake \\ \\ y: \ represents \ the \ number \ of \ frogs \ in \ the \ lake`

As you can see,
`x` is in the denominator, therefore
`x\\eq 0`. Since we need to provide a reasonable domain, we say that the domain is the set of natural numbers and this doesn't include the number 0. Why aren't negative values included in the domain as well? Well, although negative values are included in the function, they aren't reasonable because
`x` represents the number of snakes and you always get positive numbers when counting things.

To get the range, let's take the inverse of this function, so:

`y=f(x)=(100)/(x) \\ \\ \\ Interchange \ x \ and \ y: \\ \\ x=(100)/(y) \\ \\ \\ Isolate \ y: \\ \\ y=(100)/(x) \\ \\ f^(-1)(x)=(100)/(x)`

So the domain of
`f^(-1)(x)` is the range of our given function
`y=f(x)`. As you can see the inverse function is the same as our given function, then the range is the set of natural numbers as well.

2. Graph.

First of all, we can define the pattern of the rational function as:

`y=g(x)=(1)/(x)`

So our function will be:

`f(x)=100g(x)=100(1)/(x)=(100)/(x)`

So the graph of
`f(x)` will be the same graph of
`g(x)` but it's been stretched vertically by a constant of 100, that is, each y-value is multiplied by 100. Also, since
`x \\eq 0` and
`y \\eq 0` then at
`x=0` there is a vertical asymptote and at
`y=0` there is an horizontal asymptote. Finally, the graph is shown below for
`x>0 \ and \ y>0`, but remember: Whenever
`x \ and \ y` are natural numbers.