Graphing Inverse Tangent and Identifying the Domain and Range

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In this discussion, we are going to graph inverse tangent and identify the domain and range.

Key things to consider:
(skip ahead if you like)

🔷Begin with the Graph of the Tangent Function
(skip to 0:08)

🔷Restrict the Domain (-pi/2, pi/2)
(skip to 1:11)

🔷To Graph Inverse tangent, do the Following:
(skip to 2:03)

🔹Step1: Draw a Number Quadrant
(skip to 2:09)

🔹Step 2: Draw the Line y = x
(skip to 2:44)

🔹Step 3: Draw the Restricted Graph of Tangent
(skip to 3:05)

🔹Step 4: Swap the x and y Values
(skip to 4:07)

🔹Step5: Reflect the New Graph about the Line y = x
(skip to 5:02 )

🔷Observe the Domain and Range of Inverse Tangent
(skip to 6:28)

Let us begin!

🔷Graph the Tangent Function

We 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 0 and pi
asymptotes at -pi/2, pi/2, and 3pi/2.

This pattern continues on and on to the right and 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


🔹Step 1: Draw a Number Quadrant

🔹Step 2: 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

🔹Step 4: Swap the x and y Values

On the original tangent graph, we have:
the point (0,0)
a vertical asymptote at x = -pi/2
an 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, which is inverse tangent, 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.

Now you are prepared to evaluate inverse tangent, which is our next discussion!

Thank you for watching! I hope that my discussion helps you on your math journey!

Sincerely,

➕ MathAngel369 ➕



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🔑 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!

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