In this discussion, we graph inverse tangent and identify the domain and range.
Key things to consider:
- Begin with the Graph of the Tangent Function
- Restrict the Domain (-pi/2 , pi/2)
- To Graph Inverse tangent, do the Following:
- Step1: Draw a Number Quadrant
- Step 2: Draw the Line y = x
- Step 3: Draw the Restricted Graph of Tangent
- 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 Tangent
Let us begin!
Graph the Tangent Function
We first observe the graph of tangent.
Starting from zero, and traveling to the right, we know that:
tan(0) = 0
tan(pi/2) is undefined.
tan(pi) = 0
tan(3pi/2) is undefined.
Starting from zero, and traveling to the left, we know that:
tan(0) = 0
tan(-pi/2) is undefined.
We draw the graph of tangent;
vertex at zero and pi
asymptotes at -pi/2, pi/2, and 3pi/2.
This pattern continues to the right and to the left.
Now we know that tangent 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 tangent function fails the horizontal line test!
Restrict the Domain from -pi/2 to pi/2
However, mathematicians are clever!
Mathematicians restricted the domain to the interval
(-pi/2, pi/2).
Now, the graph passes the horizontal line test.
Thus, tangent has an inverse!
Let us focus on the interval (-pi/2 , pi/2).
Steps to Graphing Inverse Tangent
Here is a visual example; otherwise, content continues below.
Step 1: Draw a Number Quadrant
To draw the graph of inverse tangent, we draw a neat number quadrant first.
Step 2: Draw the Line y = x
Second, draw the line y = x.
I use the points (0,0), (pi/2,pi/2), and (-pi/2,-pi/2) and then draw a dotted line through the points.
Step 3: Draw the Restricted Graph of Tangent
Third, draw the restricted graph of tangent.
Step 4: Swap the x and y Values
Fourth, swap the x and y
values.
On the original tangent graph, we have:
the point (0,0)
a vertical asymptote at x = -pi/2
a vertical asymptote at x = pi/2
When swapping the x and y values, we now have
the point (0,0)
a horizontal asymptote at y = -pi/2
a horizontal asymptote at y = pi/2
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 tangent.
Observe the Domain and Range of Inverse Tangent
Now we can identify the domain and range of inverse tangent.
The domain of inverse tangent is (-inf,inf).
This means that, if you have a function in the form y = tan^-1(x), our x-value must fall within the domain of (-inf,inf).
The Range of inverse tangent is (-pi/2,pi/2).
This means that, if you have a function in the form y = tan^-1(x), our y-value must fall within the range of (-pi/2,pi/2).
Keep in mind that we have a set of parentheses.
So, we are not actually including -pi/2 or pi/2, but we can have values that come close to them.
On a circle, we only choose values from the right region when considering the range of inverse tangent.
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Now you are prepared to evaluate inverse tangent, which is our next discussion!
Are you ready? Go to the next article!
Inverse Tangent (ArcTan) | How to Find the Exact Value
Need more example? Here are suggested articles:
Graphing Inverse Sine (ArcSine) and Identifying the Domain and Range
Graphing Inverse Cosine (ArcCosine) and Identifying the Domain and Range
Graphing Inverse Cotangent (ArcCotangent) 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!
Note: If you are my student at the university, and you happen to see this, you do not need to take this course as you already have access to everything I provide.