April 29th, 2009 | 
Author: The elementary functions are the building blocks with which we will construct calculus, a specific roundup of all the basic functions. The basic elementary functions include constant functions, power laws, exponentials, and logarithms. These functions are listed here.
The natural logarithm and exponential use a base of the number e=2.71828… which will be calculated later on. These functions can be written as $latex e^x&s=1$ and $latex \ln x &s=1$.
The exponentials and logarithms are called transcendental functions because they can’t be calculated exactly in a finite number of steps for every input. As an example ln 2 can be expressed as
There are an infinite number of terms that need to be added up to compute the value. Transcendental is a complex concept and we won’t be using much of it in our introductory studies.
The key concept of elementary functions is that they can be combined with the basic arithmetic operations, exponentiation, or composition to create a new elementary function. Some common sets of elementary functions are the trigonometric and hyperbolic functions. There inverse functions are also elementary. The trigonometric functions $latex \sin x &s=1$, $latex \cos x &s=1$, $latex \tan x &s=1$, $latex \sec x &s=1$, $latex \cot x &s=1$, and $latex \csc x &s=1$.
The inverse trigonometric functions are $latex \text{arcsin} x&s=1$, $latex \text{arccos}\, x &s=1$, $latex \text{arctan}\, x &s=1$, $latex \text{arcsec}\, x &s=1$, and $latex \text{arccot}\, x &s=1$, $latex \text{arccsc}\, x &s=1$.
The hyperbolic functions are $latex \sinh x&s=1$, $latex \cosh x&s=1$, $latex \tanh x&s=1$, $latex \text{sech}\, x&s=1$, $latex \text{coth}\, x&s=1$, and $latex \text{csch}\, x&s=1$.
The inverse hyperbolic functions are $latex \text{arcsinh}\, x&s=1$, $latex \text{arccosh}\, x&s=1$, $latex \text{arctanh}\, x&s=1$, $latex \text{arcsech}\, x &s=1$, $latex \text{arccoth}\, x &s=1$, and $latex \text{arccsch}\, x &s=1$.
Any elementary function can be entered into a standard scientific calculator and computed. How to program the calculator to calculate these quantities is one objective of calculus.
In general, it gives more understanding if the trigonometric and hyperbolic functions are expressed in terms of complex functions. We will only use the values along the real line in introductory calculus, but the more general definitions allow you to explore the structures amongst the functions in more advanced calculus.
The complex trigonometric functions
The complex hyperbolic functions
The inverse trigonometric functions
![\begin{array}{ll} \text{arcsin}\,z = -i\ln\left(iz+\sqrt{1-z^2}\right) &\quad \text{arccos}\,z =\pi/2 +i\ln\left(iz+\sqrt{1-z^2}\right)\\ \\ \text{arcsec}\, z = \pi/2 +i\ln\left(\sqrt{1-\cfrac{1}{z^2}} +\cfrac{i}{z}\right) &\quad \text{arccsc}\, z = -i\ln\left(\sqrt{1-\cfrac{1}{z^2}} +\cfrac{i}{z}\right) \\ \\ \text{arctan}\,z =\cfrac{i}{2}\ln\left(\cfrac{1-iz}{1+iz}\right)&\quad \text{arccot}\,z =\cfrac{i}{2}\left[\ln\left(\cfrac{z-i}{z}\right)-\ln\left(\cfrac{z+i}{z}\right)\right] \\ \\ \end{array}](http://l.wordpress.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D%20%5Ctext%7Barcsin%7D%5C%2Cz%20%3D%20-i%5Cln%5Cleft%28iz%2B%5Csqrt%7B1-z%5E2%7D%5Cright%29%20%26amp%3B%5Cquad%20%5Ctext%7Barccos%7D%5C%2Cz%20%3D%5Cpi%2F2%20%2Bi%5Cln%5Cleft%28iz%2B%5Csqrt%7B1-z%5E2%7D%5Cright%29%5C%5C%20%5C%5C%20%5Ctext%7Barcsec%7D%5C%2C%20z%20%3D%20%5Cpi%2F2%20%2Bi%5Cln%5Cleft%28%5Csqrt%7B1-%5Ccfrac%7B1%7D%7Bz%5E2%7D%7D%20%2B%5Ccfrac%7Bi%7D%7Bz%7D%5Cright%29%20%26amp%3B%5Cquad%20%5Ctext%7Barccsc%7D%5C%2C%20z%20%3D%20-i%5Cln%5Cleft%28%5Csqrt%7B1-%5Ccfrac%7B1%7D%7Bz%5E2%7D%7D%20%2B%5Ccfrac%7Bi%7D%7Bz%7D%5Cright%29%20%5C%5C%20%5C%5C%20%5Ctext%7Barctan%7D%5C%2Cz%20%3D%5Ccfrac%7Bi%7D%7B2%7D%5Cln%5Cleft%28%5Ccfrac%7B1-iz%7D%7B1%2Biz%7D%5Cright%29%26amp%3B%5Cquad%20%5Ctext%7Barccot%7D%5C%2Cz%20%3D%5Ccfrac%7Bi%7D%7B2%7D%5Cleft%5B%5Cln%5Cleft%28%5Ccfrac%7Bz-i%7D%7Bz%7D%5Cright%29-%5Cln%5Cleft%28%5Ccfrac%7Bz%2Bi%7D%7Bz%7D%5Cright%29%5Cright%5D%20%5C%5C%20%5C%5C%20%5Cend%7Barray%7D&bg=FFFFFF&fg=000000&s=1 "\begin{array}{ll} \text{arcsin}\,z = -i\ln\left(iz+\sqrt{1-z^2}\right) &\quad \text{arccos}\,z =\pi/2 +i\ln\left(iz+\sqrt{1-z^2}\right)\\ \\ \text{arcsec}\, z = \pi/2 +i\ln\left(\sqrt{1-\cfrac{1}{z^2}} +\cfrac{i}{z}\right) &\quad \text{arccsc}\, z = -i\ln\left(\sqrt{1-\cfrac{1}{z^2}} +\cfrac{i}{z}\right) \\ \\ \text{arctan}\,z =\cfrac{i}{2}\ln\left(\cfrac{1-iz}{1+iz}\right)&\quad \text{arccot}\,z =\cfrac{i}{2}\left[\ln\left(\cfrac{z-i}{z}\right)-\ln\left(\cfrac{z+i}{z}\right)\right] \\ \\ \end{array}")
The inverse hyperbolic functions
![\begin{array}{ll} \text{arcsinh}\,z = \ln\left( z+\sqrt{z^2+1} \right) &\quad \text{arccosh}\,z = \ln\left(z+\sqrt{z-1}\sqrt{z+1} \right) \\ \\ \text{arccsch}\,z = \ln\left(\sqrt{1+\cfrac{1}{z^2}} + \cfrac{1}{z} \right)&\quad \text{arcsech}\,z = \ln\left( \sqrt{\cfrac{1}{z}-1} \sqrt{1+\cfrac{1} {z}} + \cfrac{1}{z} \right)\\ \\ \text{arctanh}\,z = \cfrac{1} {2} \left[\ln (1+z)-\ln (1-z)\right] &\quad \text{arccoth}\,z = \cfrac{1} {2} \left[ \ln \left( 1+\cfrac{1}{z} \right) - \ln \left( 1-\cfrac{1}{z} \right)\right] \\ \\ \end{array}](http://l.wordpress.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D%20%5Ctext%7Barcsinh%7D%5C%2Cz%20%3D%20%5Cln%5Cleft%28%20z%2B%5Csqrt%7Bz%5E2%2B1%7D%20%5Cright%29%20%26amp%3B%5Cquad%20%5Ctext%7Barccosh%7D%5C%2Cz%20%3D%20%5Cln%5Cleft%28z%2B%5Csqrt%7Bz-1%7D%5Csqrt%7Bz%2B1%7D%20%5Cright%29%20%5C%5C%20%5C%5C%20%5Ctext%7Barccsch%7D%5C%2Cz%20%3D%20%5Cln%5Cleft%28%5Csqrt%7B1%2B%5Ccfrac%7B1%7D%7Bz%5E2%7D%7D%20%2B%20%5Ccfrac%7B1%7D%7Bz%7D%20%5Cright%29%26amp%3B%5Cquad%20%5Ctext%7Barcsech%7D%5C%2Cz%20%3D%20%5Cln%5Cleft%28%20%5Csqrt%7B%5Ccfrac%7B1%7D%7Bz%7D-1%7D%20%5Csqrt%7B1%2B%5Ccfrac%7B1%7D%20%7Bz%7D%7D%20%2B%20%5Ccfrac%7B1%7D%7Bz%7D%20%5Cright%29%5C%5C%20%5C%5C%20%5Ctext%7Barctanh%7D%5C%2Cz%20%3D%20%5Ccfrac%7B1%7D%20%7B2%7D%20%5Cleft%5B%5Cln%20%281%2Bz%29-%5Cln%20%281-z%29%5Cright%5D%20%26amp%3B%5Cquad%20%5Ctext%7Barccoth%7D%5C%2Cz%20%3D%20%5Ccfrac%7B1%7D%20%7B2%7D%20%5Cleft%5B%20%5Cln%20%5Cleft%28%201%2B%5Ccfrac%7B1%7D%7Bz%7D%20%5Cright%29%20-%20%5Cln%20%5Cleft%28%201-%5Ccfrac%7B1%7D%7Bz%7D%20%5Cright%29%5Cright%5D%20%5C%5C%20%5C%5C%20%5Cend%7Barray%7D&bg=FFFFFF&fg=000000&s=1 "\begin{array}{ll} \text{arcsinh}\,z = \ln\left( z+\sqrt{z^2+1} \right) &\quad \text{arccosh}\,z = \ln\left(z+\sqrt{z-1}\sqrt{z+1} \right) \\ \\ \text{arccsch}\,z = \ln\left(\sqrt{1+\cfrac{1}{z^2}} + \cfrac{1}{z} \right)&\quad \text{arcsech}\,z = \ln\left( \sqrt{\cfrac{1}{z}-1} \sqrt{1+\cfrac{1} {z}} + \cfrac{1}{z} \right)\\ \\ \text{arctanh}\,z = \cfrac{1} {2} \left[\ln (1+z)-\ln (1-z)\right] &\quad \text{arccoth}\,z = \cfrac{1} {2} \left[ \ln \left( 1+\cfrac{1}{z} \right) - \ln \left( 1-\cfrac{1}{z} \right)\right] \\ \\ \end{array}")
We will work primarily we real functions. To express these functions in terms of real functions we replace z with x and there are also some restrictions on which x can be entered into the function. You have to use points in the real domain. Even though some of these functions have complex numbers in them but they come out real. For example, take at the value x=1
This gives the right result which we know is correct. Another troublesome detail about working with complex functions is that they can be multi-valued like for $latex \ln z&s=1$. We know that $latex 1,e^{2\pi i}, e^{4\pi i},..&s=1$ are the same number but when you take the natural logarithm of them you get different results
We have to make further restrictions on complex functions to pick a particular “branch”. Perhaps this is one of the many reasons complex variables are avoided when learning calculus. I think since you have probably gone to some effort to learn about complex numbers we should use them in learning calculus, well at least just a little bit. Complex analysis can be quite powerful in calculus.
When working with complex numbers you can’t take for granted certain calculations that work with real number but not with complex numbers. For example for real numbers we might write $latex \sqrt{-a}\sqrt{-b}= \sqrt{ab}&s=1$ this is actually not correct since for example $latex \sqrt{-1} \sqrt{-1} = i^2=-1&s=1$ but if you just combined them you would get $latex \sqrt{-1}\sqrt{-1}=\sqrt{1}=1&s=1$. So if you are doing a calculation with complex numbers I would be very careful how you entered the calculation into a calculator or do it by hand. Do each piece exactly as it is written and don’t try too many shortcuts unless you know what you are doing.
Another common simplification that is taken for granted is when you take a square root you always have to include your absolute value signs unless the result is a perfect square. For example,
So there is one expression for the term when $latex x>0 &s=1$ and another when $latex x<0&s=1$. These types of expressions are present in the formulas listed above. Getting absolute value signs correct is part of the technical details of doing calculus properly.
Now we will list the domains and ranges of the trigonometric functions, hyperbolic functions, and their inverses. If you have the inputs and definition of the functions in terms of an equation you are in pretty good shape to do calculations with them.
\noindent The domain and range of the trigonometric functions for real inputs.
![\begin{array}{lll}\text{Function} &\qquad \text{Domain} &\qquad \text{Range}\\\sin x &\qquad x \in (-\infty, \infty) &\qquad [-1,1]\\\cos x &\qquad x \in (-\infty, \infty) &\qquad [-1,1]\\\tan x &\qquad x \in (-\infty, \infty)/\{(\mathbb{Z}+1/2)\pi\} &\qquad (-\infty,\infty)\\\sec x &\qquad x \in (-\infty, \infty)/\{(\mathbb{Z}+1/2)\pi\} &\qquad (-\infty,1]\cup[1,\infty)\\\cot x &\qquad x \in (-\infty, \infty)/\{\mathbb{Z}\pi\} &\qquad (-\infty,\infty)\\\csc x &\qquad x \in (-\infty, \infty)/\{\mathbb{Z}\pi\} &\qquad (-\infty,1]\cup[1,\infty)\\\end{array}](http://l.wordpress.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D%5Ctext%7BFunction%7D%20%26amp%3B%5Cqquad%20%5Ctext%7BDomain%7D%20%26amp%3B%5Cqquad%20%5Ctext%7BRange%7D%5C%5C%5Csin%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%5B-1%2C1%5D%5C%5C%5Ccos%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%5B-1%2C1%5D%5C%5C%5Ctan%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%2F%5C%7B%28%5Cmathbb%7BZ%7D%2B1%2F2%29%5Cpi%5C%7D%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C%5Cinfty%29%5C%5C%5Csec%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%2F%5C%7B%28%5Cmathbb%7BZ%7D%2B1%2F2%29%5Cpi%5C%7D%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C1%5D%5Ccup%5B1%2C%5Cinfty%29%5C%5C%5Ccot%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%2F%5C%7B%5Cmathbb%7BZ%7D%5Cpi%5C%7D%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C%5Cinfty%29%5C%5C%5Ccsc%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%2F%5C%7B%5Cmathbb%7BZ%7D%5Cpi%5C%7D%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C1%5D%5Ccup%5B1%2C%5Cinfty%29%5C%5C%5Cend%7Barray%7D%20&bg=FFFFFF&fg=000000&s=1 "\begin{array}{lll}\text{Function} &\qquad \text{Domain} &\qquad \text{Range}\\\sin x &\qquad x \in (-\infty, \infty) &\qquad [-1,1]\\\cos x &\qquad x \in (-\infty, \infty) &\qquad [-1,1]\\\tan x &\qquad x \in (-\infty, \infty)/\{(\mathbb{Z}+1/2)\pi\} &\qquad (-\infty,\infty)\\\sec x &\qquad x \in (-\infty, \infty)/\{(\mathbb{Z}+1/2)\pi\} &\qquad (-\infty,1]\cup[1,\infty)\\\cot x &\qquad x \in (-\infty, \infty)/\{\mathbb{Z}\pi\} &\qquad (-\infty,\infty)\\\csc x &\qquad x \in (-\infty, \infty)/\{\mathbb{Z}\pi\} &\qquad (-\infty,1]\cup[1,\infty)\\\end{array}")
To get a feel for these domains and ranges we know that you can input any number into sine and cosine. The output is between -1 and 1 inclusive because their locations are on the unit circle. Tangent and secant are defined everywhere their denominator is not equal to zero. These points to be avoided are
For cotangent and cosecant the denominator cannot be zero also this occurs at the points
The range of all the functions can be seen by plotting the functions. For secant and cosecant we divide by a number that is between -1 and 1 so $latex |\sec x|\ge 1 &s=1$ and $latex |\csc x| \ge 1 &s=1$.
The domain and range of the hyperbolic functions for real inputs.
![\begin{array}{lll}\text{Function} &\qquad \text{Domain} &\qquad \text{Range}\\\text{sinh}\, x &\qquad x \in (-\infty, \infty) &\qquad (-\infty,\infty)\\\text{cosh}\, x &\qquad x \in (-\infty, \infty) &\qquad [1,\infty)\\\text{tanh}\, x &\qquad x \in (-\infty, \infty) &\qquad (-1,1)\\\text{coth}\, x &\qquad x \in (-\infty,0)\cup(0, \infty) &\qquad (-\infty,0)\cup(0,\infty)\\\text{sech}\, x &\qquad x \in (-\infty, \infty) &\qquad (0,1]\\\text{csch}\, x &\qquad x \in (-\infty,0)\cup(0, \infty) &\qquad (-\infty,0)\cup(0,\infty)\\\end{array}](http://l.wordpress.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D%5Ctext%7BFunction%7D%20%26amp%3B%5Cqquad%20%5Ctext%7BDomain%7D%20%26amp%3B%5Cqquad%20%5Ctext%7BRange%7D%5C%5C%5Ctext%7Bsinh%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C%5Cinfty%29%5C%5C%5Ctext%7Bcosh%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%5B1%2C%5Cinfty%29%5C%5C%5Ctext%7Btanh%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%28-1%2C1%29%5C%5C%5Ctext%7Bcoth%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C0%29%5Ccup%280%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C0%29%5Ccup%280%2C%5Cinfty%29%5C%5C%5Ctext%7Bsech%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%280%2C1%5D%5C%5C%5Ctext%7Bcsch%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C0%29%5Ccup%280%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C0%29%5Ccup%280%2C%5Cinfty%29%5C%5C%5Cend%7Barray%7D%20&bg=FFFFFF&fg=000000&s=1 "\begin{array}{lll}\text{Function} &\qquad \text{Domain} &\qquad \text{Range}\\\text{sinh}\, x &\qquad x \in (-\infty, \infty) &\qquad (-\infty,\infty)\\\text{cosh}\, x &\qquad x \in (-\infty, \infty) &\qquad [1,\infty)\\\text{tanh}\, x &\qquad x \in (-\infty, \infty) &\qquad (-1,1)\\\text{coth}\, x &\qquad x \in (-\infty,0)\cup(0, \infty) &\qquad (-\infty,0)\cup(0,\infty)\\\text{sech}\, x &\qquad x \in (-\infty, \infty) &\qquad (0,1]\\\text{csch}\, x &\qquad x \in (-\infty,0)\cup(0, \infty) &\qquad (-\infty,0)\cup(0,\infty)\\\end{array}")
The hyperbolic functions are constructed from $latex \sinh x&s=1$ and $latex \cosh x&s=1$ so we look to these functions for our starting information. $latex \cosh x&s=1$ is always positive while $latex \sinh x&s=1$ is both positive and negative and crosses zero at exactly one point $latex x=0&s=1$. For the functions where the denominator is $latex \sinh x&s=1$ we must exclude $latex x=0&s=1$ from the domain. These two cases are $latex \text{coth}\,x&s=1$ and $latex\text{csch}\,x&s=1$. For the other two hyperbolic functions $latex \sinh x&s=1$ is not present in the denominator so they have the full range of the real line. Since $latex \cosh x \ge 1&s=1$ then $latex \text{sech}\, x&s=1$ is in (0,1]. For the other functions since the numerator is never zero then these functions have the value 0 excluded from the range.
The domain and range of the inverse trigonometric functions for real inputs.
![\begin{array}{lll}\text{Function} &\qquad \text{Domain} &\qquad \text{Range}\\\text{arcsin}\, x &\qquad x \in [-1,1] &\qquad [-\pi/2,\pi/2]\\\text{arccos}\, x &\qquad x \in [-1,1] &\qquad [0,\pi]\\\text{arctan}\, x &\qquad x \in (-\infty, \infty) &\qquad (-\pi/2,\pi/2)\\\text{arccot}\, x &\qquad x \in (-\infty, \infty) &\qquad (0,\pi)\\\text{arcsec}\, x &\qquad x \in (-\infty,-1)\cup(1,\infty) &\qquad (0,\pi)/\{\pi/2\}\\\text{arccsc}\, x &\qquad x \in (-\infty,-1)\cup(1,\infty) &\qquad [-\pi/2,0)\cup(0,\pi/2]\\\end{array}](http://l.wordpress.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D%5Ctext%7BFunction%7D%20%26amp%3B%5Cqquad%20%5Ctext%7BDomain%7D%20%26amp%3B%5Cqquad%20%5Ctext%7BRange%7D%5C%5C%5Ctext%7Barcsin%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%5B-1%2C1%5D%20%26amp%3B%5Cqquad%20%5B-%5Cpi%2F2%2C%5Cpi%2F2%5D%5C%5C%5Ctext%7Barccos%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%5B-1%2C1%5D%20%26amp%3B%5Cqquad%20%5B0%2C%5Cpi%5D%5C%5C%5Ctext%7Barctan%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%28-%5Cpi%2F2%2C%5Cpi%2F2%29%5C%5C%5Ctext%7Barccot%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%280%2C%5Cpi%29%5C%5C%5Ctext%7Barcsec%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C-1%29%5Ccup%281%2C%5Cinfty%29%20%26amp%3B%5Cqquad%20%280%2C%5Cpi%29%2F%5C%7B%5Cpi%2F2%5C%7D%5C%5C%5Ctext%7Barccsc%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C-1%29%5Ccup%281%2C%5Cinfty%29%20%26amp%3B%5Cqquad%20%5B-%5Cpi%2F2%2C0%29%5Ccup%280%2C%5Cpi%2F2%5D%5C%5C%5Cend%7Barray%7D%20&bg=FFFFFF&fg=000000&s=1 "\begin{array}{lll}\text{Function} &\qquad \text{Domain} &\qquad \text{Range}\\\text{arcsin}\, x &\qquad x \in [-1,1] &\qquad [-\pi/2,\pi/2]\\\text{arccos}\, x &\qquad x \in [-1,1] &\qquad [0,\pi]\\\text{arctan}\, x &\qquad x \in (-\infty, \infty) &\qquad (-\pi/2,\pi/2)\\\text{arccot}\, x &\qquad x \in (-\infty, \infty) &\qquad (0,\pi)\\\text{arcsec}\, x &\qquad x \in (-\infty,-1)\cup(1,\infty) &\qquad (0,\pi)/\{\pi/2\}\\\text{arccsc}\, x &\qquad x \in (-\infty,-1)\cup(1,\infty) &\qquad [-\pi/2,0)\cup(0,\pi/2]\\\end{array}")
One approach to studying the inverse trigonometric functions is to plot them as the inverse functions of the normal trigonometric functions. Another approach is to just define them as we have done above. One can think of the domain and range of the inverse function exchanging with the normal function. By this method we understand the domain of all these functions. The range of the inverse functions are indicative of how the domain of the regular functions were restricted for example look at the domain of $latex \text{arcsin} x&s=1$ and $latex \text{arccos} x&s=1$.
The domain and range of the inverse hyperbolic functions for real inputs.
![\begin{array}{lll}\text{Function} &\qquad \text{Domain} &\qquad \text{Range}\\\text{arcsinh}\, x &\qquad x \in (-\infty, \infty) &\qquad (-\infty,\infty)\\\text{arccosh}\, x &\qquad x \in [1, \infty) &\qquad [0,\infty)\\\text{arctanh}\, x &\qquad x \in (-1, 1) &\qquad (-\infty,\infty)\\\text{arccoth}\, x &\qquad x \in (-\infty,-1)\cup(1,\infty) &\qquad (-\infty,\infty)\\\text{arcsech}\, x &\qquad x \in (0,1] &\qquad [0,\infty)\\\text{arccsch}\, x &\qquad x \in (-\infty, \infty) &\qquad (-\infty,\infty)\\\end{array}](http://l.wordpress.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D%5Ctext%7BFunction%7D%20%26amp%3B%5Cqquad%20%5Ctext%7BDomain%7D%20%26amp%3B%5Cqquad%20%5Ctext%7BRange%7D%5C%5C%5Ctext%7Barcsinh%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C%5Cinfty%29%5C%5C%5Ctext%7Barccosh%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%5B1%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%5B0%2C%5Cinfty%29%5C%5C%5Ctext%7Barctanh%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-1%2C%201%29%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C%5Cinfty%29%5C%5C%5Ctext%7Barccoth%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C-1%29%5Ccup%281%2C%5Cinfty%29%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C%5Cinfty%29%5C%5C%5Ctext%7Barcsech%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%280%2C1%5D%20%26amp%3B%5Cqquad%20%5B0%2C%5Cinfty%29%5C%5C%5Ctext%7Barccsch%7D%5C%2C%20x%20%26amp%3B%5Cqquad%20x%20%5Cin%20%28-%5Cinfty%2C%20%5Cinfty%29%20%26amp%3B%5Cqquad%20%28-%5Cinfty%2C%5Cinfty%29%5C%5C%5Cend%7Barray%7D%20&bg=FFFFFF&fg=000000&s=1 "\begin{array}{lll}\text{Function} &\qquad \text{Domain} &\qquad \text{Range}\\\text{arcsinh}\, x &\qquad x \in (-\infty, \infty) &\qquad (-\infty,\infty)\\\text{arccosh}\, x &\qquad x \in [1, \infty) &\qquad [0,\infty)\\\text{arctanh}\, x &\qquad x \in (-1, 1) &\qquad (-\infty,\infty)\\\text{arccoth}\, x &\qquad x \in (-\infty,-1)\cup(1,\infty) &\qquad (-\infty,\infty)\\\text{arcsech}\, x &\qquad x \in (0,1] &\qquad [0,\infty)\\\text{arccsch}\, x &\qquad x \in (-\infty, \infty) &\qquad (-\infty,\infty)\\\end{array}")
The inverse hyperbolic functions are quite foreign. You can understand their shapes by comparing them to the regular functions mirrored along $y=x$, you can work with the definitions above, or simply just plot them.
What I call the basic elementary functions are all the functions I have listed above. Elementary functions can be constructed from the basic elementary functions through any operation of +,-,/,*, or composition. For example, $latex f(x)=mx + b &s=1$ is an elementary function, we take a constant function multiply it by another power function and add another constant. Polynomials are important elementary functions which are just the sum of power laws with constant coefficients.
The ratio of two polynomials is an elementary function also called rational functions.
Since power laws can be fractional we have other elementary functions such as
Composition with the trigonometric or hyperbolic functions is also possible to give another elementary functions.
As you can see there are infinite possibilities. The elementary functions are important because as we study calculus we use them over and over again. Also certain calculations on the elementary functions is possible in terms of results that are themselves elementary functions. Elementary functions can also be represented as the solution to a differential equation something we learn after differentiation and integration.
An example of a function that is not elementary is the factorial function. This function belongs to a larger class of functions called special functions. Elementary functions are a subset of the special functions. Special functions are what you learn after you have mastered calculus and go hunting for more functions.
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