Graph, Domain and Range of Arctan(x) Function
The graph, domain, range, and other properties of the inverse trigonometric function \( \arctan(x) \) are explored using graphs, examples with detailed solutions, and an interactive app.
Definition of arctan(x) Functions
Examining the graph of \( \tan(x) \), shown below, we note that it is not a one-to-one function on its implied domain. But if we limit the domain to \( \left(-\dfrac{\pi}{2} , \dfrac{\pi}{2}\right) \) (blue graph below), we obtain a one-to-one function that has an inverse which cannot be obtained algebraically.
.gif)
The inverse function of \( f(x) = \tan(x) , x \in \left(-\dfrac{\pi}{2} , \dfrac{\pi}{2}\right) \) is \( f^{-1} = \arctan(x) \).
We define \( \arctan(x) \) as follows:
\[ y = \arctan(x) \iff x = \tan(y) \]
where \( x \in (-\infty , +\infty) \) and \( y \in \left(-\dfrac{\pi}{2} , \dfrac{\pi}{2}\right) \).
Let us make a table of values of \( y = \arctan(x) \) and graph it along with \( y = \tan(x), x \in (-\infty , +\infty) \).
| \( x \) | \(-10000\) | \(-100\) | \(-10\) | \(-1\) | \(0\) | \(1\) | \(10\) | \(100\) | \(10000\) |
| \( y = \arctan(x) \) | \(-1.57069\) | \(-1.56079\) | \(-1.47112\) | \(-\dfrac{\pi}{4}\) | \(0\) | \( \dfrac{\pi}{4} \) | \(1.47112\) | \(1.56079\) | \(1.57069\) |
| Reason | calculator | calculator | calculator | since \( \tan\left(-\dfrac{\pi}{4}\right) = -1 \) | since \( \tan(0) = 0 \) | since \( \tan\left(\dfrac{\pi}{4}\right) = 1 \) | calculator | calculator | calculator |
The graphs of \( y = \arctan(x) \) and \( y = \tan(x) , x \in \left(-\dfrac{\pi}{2} , \dfrac{\pi}{2}\right) \) are shown below. Being inverses of each other, each of the two graphs is the reflection of the other across the line \( y = x \).
Note: Because \( \tan(x) \) has vertical asymptotes at \( x = -\dfrac{\pi}{2} \) and \( x = \dfrac{\pi}{2} \) (blue broken lines), its inverse function \( \arctan(x) \) has horizontal asymptotes (red broken lines).
The behavior of the graph of \( \arctan(x) \) close to the horizontal asymptotes may be described using the concept of limits as follows:
\[ \lim_{x\to\infty} \arctan(x) = \dfrac{\pi}{2} \]
and
\[ \lim_{x\to -\infty} \arctan(x) = -\dfrac{\pi}{2} \]
-and-arctan(x).gif)
Examples & Step-by-Step Solutions
Example 1: Evaluating Arctan Values
Problem: Evaluate \( \arctan(x) \) given the value of \( x \).
Solution:
Special values related to special angles:
- \( \arctan(0) = 0 \) because \( \tan(0) = 0 \)
- \( \arctan(-1) = -\dfrac{\pi}{4} \; \text{ or } -45^{\circ} \) because \( \tan\left(-\dfrac{\pi}{4}\right) = -1 \)
- \( \arctan(1) = \dfrac{\pi}{4} \; \text{ or } 45^{\circ} \) because \( \tan\left(\dfrac{\pi}{4}\right) = 1 \)
Use of calculator:
- \( \arctan(-2) \approx -1.107 \; \text{ rad or } -63.43^{\circ} \)
- \( \arctan(180) \approx 1.56 \; \text{ rad or } 89.68^{\circ} \)
- \( \arctan(-230) \approx -1.566 \; \text{ rad or } -89.75^{\circ} \)
- \( \arctan(-0.2) \approx -0.197 \; \text{ rad or } -11.31^{\circ} \)
Properties of \( y = \arctan(x) \)
- Domain: \( (-\infty , +\infty) \)
- Range: \( \left(-\dfrac{\pi}{2} , \dfrac{\pi}{2}\right) \)
- \( \arctan(-x) = - \arctan(x) \), hence \( \arctan(x) \) is an odd function.
- \( \arctan(x) \) is a one-to-one function.
- \( \tan(\arctan(x)) = x \), for all real \( x \), due to the property of a function and its inverse: \( f(f^{-1}(x)) = x \).
- \( \arctan(\tan(x)) = x \), for \( x \) in the interval \( \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right) \), due to the property of a function and its inverse: \( f^{-1}(f(x)) = x \).
Example 2: Finding the Range of Transformed Arctan Functions
Problem: Find the range of the functions:
a) \( y = 3 \arctan(x) \)
b) \( y = - \arctan(x) + \pi/2 \)
c) \( y = 2 \arctan(x + 3) - \pi/4 \)
Solution:
-
a) Start with the range of \( \arctan(x) \):
\[ -\dfrac{\pi}{2} \lt \arctan(x) \lt \dfrac{\pi}{2} \]
Multiply all terms by \( 3 \):
\[ -\dfrac{3\pi}{2} \lt 3 \arctan(x) \lt \dfrac{3\pi}{2} \]
The range is \( \left(-\dfrac{3\pi}{2}, \dfrac{3\pi}{2}\right) \).
-
b) Start with \( -\dfrac{\pi}{2} \lt \arctan(x) \lt \dfrac{\pi}{2} \). Multiply by \( -1 \) and reverse the inequality signs:
\[ -\dfrac{\pi}{2} \lt -\arctan(x) \lt \dfrac{\pi}{2} \]
Add \( \dfrac{\pi}{2} \) to all terms:
\[ 0 \lt - \arctan(x) + \dfrac{\pi}{2} \lt \pi \]
The range is \( (0, \pi) \).
-
c) The graph of \( y = \arctan(x+3) \) is shifted 3 units to the left, which does not affect the range \( \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right) \). Thus:
\[ -\dfrac{\pi}{2} \lt \arctan(x+3) \lt \dfrac{\pi}{2} \]
Multiply all terms by \( 2 \):
\[ -\pi \lt 2 \arctan(x+3) \lt \pi \]
Subtract \( \dfrac{\pi}{4} \) from all terms:
\[ -\pi - \dfrac{\pi}{4} \lt 2 \arctan(x+3) - \dfrac{\pi}{4} \lt \pi - \dfrac{\pi}{4} \implies -\dfrac{5\pi}{4} \lt y \lt \dfrac{3\pi}{4} \]
The range is \( \left(-\dfrac{5\pi}{4}, \dfrac{3\pi}{4}\right) \).
Example 3: Evaluating Composite Expressions
Problem: Evaluate if possible:
a) \( \tan(\arctan(-1.5)) \)
b) \( \arctan\left(\tan\left(\dfrac{\pi}{7}\right)\right) \)
c) \( \arctan\left(\tan\left(-\dfrac{\pi}{2}\right)\right) \)
d) \( \tan(\arctan(-19.5)) \)
e) \( \arctan\left(\tan\left(\dfrac{13\pi}{4}\right)\right) \)
Solution:
- a) \( \tan(\arctan(-1.5)) = -1.5 \) using property 5.
- b) \( \arctan\left(\tan\left(\dfrac{\pi}{7}\right)\right) = \dfrac{\pi}{7} \) using property 6.
- c) Property 6 cannot be used because \( \tan\left(-\dfrac{\pi}{2}\right) \) is undefined. Therefore, \( \arctan\left(\tan\left(-\dfrac{\pi}{2}\right)\right) \) is undefined.
- d) \( \tan(\arctan(-19.5)) = -19.5 \) using property 5.
- e) Property 6 cannot be directly applied because \( \dfrac{13\pi}{4} \) is outside the interval \( \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right) \). First, evaluate:
\[ \tan\left(\dfrac{13\pi}{4}\right) = \tan\left(\dfrac{12\pi}{4} + \dfrac{\pi}{4}\right) = \tan\left(3\pi + \dfrac{\pi}{4}\right) = \tan\left(\dfrac{\pi}{4}\right) = 1 \]
Substitute this back into the expression:
\[ \arctan\left(\tan\left(\dfrac{13\pi}{4}\right)\right) = \arctan(1) = \dfrac{\pi}{4} \]
Interactive Tutorial to Explore the Transformed arctan(x)
The exploration is carried out by analyzing the effects of the coefficients \( a, b, c \) and \( d \) included in the more general arctan function given by:
\[ f(x) = a \arctan(b x + c) + d \]
Change coefficients \( a, b, c \) and \( d \) and click on the button 'draw' in the left panel below. Zoom in and out for better viewing. Depending on which coefficients are changing, we expect the graph to be transformed through vertical and horizontal compression, stretching, reflection, and also vertical shifting.
- Set the coefficients to \( a = 1, b = 1, c = 0 \) and \( d = 0 \) to obtain:
\[ f(x) = \arctan(x) \]
Check that the domain of \( \arctan(x) \) is the set of all real numbers and the range is given by the interval \( \left(-\dfrac{\pi}{2} , \dfrac{\pi}{2}\right) \). Check also that \( \arctan(x) \) has horizontal asymptotes at \( y = -\dfrac{\pi}{2} \) and \( y = \dfrac{\pi}{2} \). (Zoom in and out)
- Change coefficient \( a \) and note how the graph of \( a \arctan(x) \) changes (vertical compression, stretching, reflection). How does it affect the range and the asymptotes of the function? Does a change in coefficient \( a \) affect the domain of the function?
- Change coefficient \( b \) and note how the graph of the function changes (horizontal compression, stretching). Does a change in \( b \) affect the range, domain, and asymptotes of the function?
- Change coefficient \( c \) and note how the graph of the function changes (horizontal shift). Does a change in \( c \) affect the domain, range, and asymptotes of the function?
- Change coefficient \( d \) and note how the graph of the function changes (vertical shift). Does a change in \( d \) affect the range, domain, and asymptotes of the function?
- If the range of \( \arctan(x) \) is given by the interval \( \left(-\dfrac{\pi}{2} , \dfrac{\pi}{2}\right) \), what is the range of \( a \arctan(x) \)? What is the range of \( a \arctan(x) + d \)?
- If the horizontal asymptotes of \( \arctan(x) \) are given by the horizontal lines \( y = -\dfrac{\pi}{2} \) and \( y = \dfrac{\pi}{2} \), what are the horizontal asymptotes of \( a \arctan(x) \)?
- What are the horizontal asymptotes of \( a \arctan(x) + d \)?
More References and Links to Inverse Trigonometric Functions
Explore additional resources on inverse trigonometric functions: