In this discussion, we are going to graph inverse cotangent and identify the domain and range.
Key things to consider:
- Begin with the Graph of the Cotangent Function
- Restrict the Domain to the interval (0,pi)
- To Graph Inverse Cotangent, do the Following:
- Step1: Draw a Number Quadrant
- Step 2: Draw the Line y = x
- Step 3: Draw the Restricted Graph of Cotangent
- Step 4: Swap the x and y Values
- Step5: Reflect the New Graph about the Line y = x
- Observe the Domain and Range of Inverse Cotangent
Let us begin!
Graph the Cotangent Function
We observe the graph of cotangent.
Starting from zero, and traveling to the right, we know that:… cot(0) is undefined
cot(pi/2) = 0
cot(pi) is undefined
cot(3pi/2) = 0
cot(2pi) is undefined.
This pattern continues to the right and left.
We draw the graph of cotangent.
Vertex: pi/2, 3pi/2
Asymptotes: 0, pi, 2pi
Now we know that cotangent is a function because it passes the vertical line test.
This means that if we draw any vertical line to the graph in its upright position, it will intersect the graph only once.
However, for a function to have an inverse, it must be one-to-one.
In other words, it must pass the horizontal line test.
This means that if we draw any horizontal line to the graph in its upright position, it must touch the graph only once.
We see that the cotangent function fails the horizontal line test!
Restrict the Domain from 0 to pi
However, mathematicians are clever!
Mathematicians restricted the domain to (0,pi).
Now, the graph passes the horizontal line test.
Thus, cotangent has an inverse!
Let us focus on the interval (0, pi).
Steps to Graphing Inverse Cotangent
Here is a visual example; otherwise, content continues below.
Step 1: Draw a Number Quadrant
To draw the graph of inverse cotangent, we want to first draw a number quadrant.
Step 2: Draw the Line y = x
Second, draw the line y = x.
Step 3: Draw the Restricted Graph of Cotangent
Third, draw the restricted graph of cotangent.
Step 4: Swap the x and y Values
Fourth, swap the x and y values.
It is important
to note that, on the original cotangent graph, we have:
a vertical asymptote at x = 0
a point at (pi/2, 0)
a vertical asymptote x = pi
When swapping the x and y values, we now have:
a horizontal asymptote at y = 0
a point at (0, pi/2)
a horizontal asymptote at y = pi
Step 5: Draw the New Graph by Reflecting about the line y = x.
Last, draw the new graph by reflecting about the line y = x.
Separating the new graph from the old one, you now have the graph of inverse cotangent.
Observe the Domain and Range of Inverse Cotangent
Now we can identify the domain and range of inverse cotangent.
The domain of inverse cotangent is (-inf,inf).
This means that, if you have a function in the form y = cot^-1(x),
our x-value must fall within the domain of (-inf,inf).
The range of inverse cotangent is (0,pi).
This means that, if you have a function in the form y = cot^-1(x),
our y-value must fall within the range of (0,pi).
Keep in mind that we have a set of parentheses. So, we are not actually including 0 or pi, but we can have values that come close to them.
On a circle, we only choose values from the upper region when considering the range of inverse cotangent.
Now you are prepared to evaluate inverse cotangent, which is our next discussion!
Are you ready to evaluate inverse cotangent? Go to the next article!
Inverse Cotangent | How to Find the Exact Value
Need more examples of how to graph inverse trig functions? Here are suggested articles:
Graphing Inverse Sine and Identifying the Domain and Range
Graphing Inverse Cosine and Identifying the Domain and Range
Graphing Inverse Tangent and Identifying the Domain and Range
Graphing Inverse Secant and Identifying the Domain and Range
Graphing Inverse Cosecant and Identifying the Domain and Range
🔑 Practice is the key! If you would like more examples,
practice problems with the answers, quizzes with the answers, and more regarding inverse trig functions, consider taking my Math Course on Inverse trig functions!