In this discussion, we learn how to graph inverse secant and identify its domain and range.
Here is an outline of this discussion:
- Begin with the Graph of the Secant Function
- Restrict the Domain of Secant
- Draw Inverse Secant by doing the Following:
- Step 1: Draw a Neat Cartesian Plane
- Step 2: Draw the Line y = x
- Step 3: Draw the Restricted Graph of Secant
- Step 4: Swap the x and y Values
- Step 5: Reflect the New Graph about the Line y = x
- Observe the Domain and Range of Inverse Secant
Let us begin!
Graph the Secant Function
Begin with the original secant graph.
Starting at zero and traveling to the right, we know that:
sec(0) = 1
sec( \frac{\pi}{2}) is undefined
sec(\pi) = -1
sec( \frac{\pi}{2}) is undefined
sec(\pi) = 1
This pattern continues to the right and to the left.
Notice that secant is a function because it passes the Vertical-Line-Test.
This means that if we draw any vertical line to the graph, it intersects the graph one time.
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 any horizontal line is drawn to the graph, it must touch the graph one time only.
We see that the secant function fails the horizontal line test!
Restrict the Domain of Secant
However, mathematicians are clever!
Restrict the
domain to
[0,\frac{\pi}{2})\cup(\frac{\pi}{2}, \pi].
Now, the graph passes the
Horizontal-Line-Test.
Thus, secant has an inverse!
Let us focus on the interval [0,\frac{\pi}{2})\cup(\frac{\pi}{2}, \pi].
Steps to Graphing Inverse Secant
Step 1: Draw a Neat Cartesian Plane
To draw the graph of inverse secant, begin by drawing a neat Cartesian Plane.
It is important to note that \frac{\pi}{2} is approximately 1.6.
Thus, place \frac{\pi}{2} somewhere in between 1 and 2.
Also, \pi is approximately 3.14.
Thus, place \pi at about the number 3.
Step 2: Draw the Line y = x
Second, draw the line y = x.
Use the points
(0,0),\, (1,1),\\ (\frac{\pi}{2},\frac{\pi}{2}),\, (\pi,\pi),\\ (-1,-1),\ (-\frac{\pi}{2},-\frac{\pi}{2}),\\ (-\pi,-\pi).
Then draw a dotted line through the points.
Step 3: Draw the Restricted Graph of Secant
Third, draw the restricted graph of secant.
Step 4: Swap the x and y Values
Fourth, swap the x and y values.
The point
(0, 1)
becomes the point
(1, 0).
The vertical asymptote
x = \frac{\pi}{2}
becomes the horizontal asymptote
y = \frac{\pi}{2}.
The point
(\pi, -1)
becomes the point
(-1, \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, we have the graph of inverse secant.
Observe the Domain and Range of Inverse Secant
Now we can identify the domain and range of inverse secant.
Read the graph from left to right to observe the domain and read the graph from bottom to top to observe the range.
The domain of inverse secant is (-\infty,1]\cup[1,\infty).
This means that, given a function in the form y = sec-1(x),
the x-value must lie within the interval (-\infty,1]\cup[1,\infty).
The range of inverse secant is
[0,\frac{\pi}{2})\cup(\frac{\pi}{2}, \pi].
This means that, given a function in the form y = sec-1(x),
the y-value must lie within the interval [0,\frac{\pi}{2})\cup(\frac{\pi}{2}, \pi].
Keep in mind that there is a set of parentheses at \frac{\pi}{2},
which means that \frac{\pi}{2} is not included in the range of inverse secant.
However, we can have values that come close to it.
On the unit circle, only choose values from the upper region (not including \frac{\pi}{2}) when considering the range of inverse secant.
I hope my blog helps you!
There is more where that came from!
🎥Suggested Playlist: Inverse Trig Functions
🔑Practice
Are you ready to get some practice? Take my Inverse Trig function course!
Become an Expert in Inverse Trig Functions
One thought on “Graphing Inverse Secant and Identifying the Domain and Range”