Domains of Functions

The domain of a function is the set of values for which it is defined.  Sometimes it is easier to say where the function is not defined.  

We assume our functions are real valued.  

So values of the independent variable which would involve the square root of negative numbers and hence complex numbers for values are not allowed.  Follow that convention until you are in course or a topic involving complex-valued functions.

Domains of Basic Functions 

• Polynomials p(u) are defined for all real numbers
• Rational functions (the ratio p(u)/g(u)  of two polynomials) are defined at points where division by zero appears, that is where g(u) = 0. 
• sin(u) and cos(u) are define for all real numbers u. . 
• The ratios and reciprocals of the sin(u) and cos(u) are defined except those points where denominators would  equal zero.
• logarithms are defined only for non-negative real numbers - when functions are restricted to being real-valued.
• square roots and even roots are defined only for non-negative real numbers - when functions are restricted to being real-valued.
• odd roots are defined for all real numbers.
• Powers x^q where q = m/n and m = 2k is even and n odd is define for all real numbers.  Here  x^q =  (x^m)^1/n =  (x^1/n)^m . Moreover,    x^q = sign(x)   |x|^q 
• Powers x^q where q = m/n and m is odd  even and n even is define for all non-negative numbers. Here  x^q =  (x^m)^1/n =  (x^1/n)^m   But  both   (x^m)^1/n and  (x^1/n)^m  requires the computation of an even root of a negative number when x is negative and n is even. So is not possible for real numbers when functions are not allowed to be complex-valued.
• When q = m/n and a > 0,  a^q = exp( q ln(a)). 
• When q = m/n and n is odd, and x is non-zero,   |x|^q = exp( q ln(|x|))   (Why?) and  x^q = sign(x)|x|^q = sign(x)*exp(q*ln(|x|))   

Composition of these basic functions yield further functions.  The domain of the resulting composite function depends on the domains of the composed functions.  Basic and composite functions may then appear in sums, differences, products and quotients to provide further further functions.  Reciprocals and Quotients of functions, basic or composite or more complicated,  introduces points where the further function is not defined in order to avoid division by zerio.

Remark: A finer discussion of the connections between powers, logarithms and exponentials is possible.  In a second course on calculus, one can show that that the properties of natural logarithms and exponential function  imply  f(x) = exp(x ln(a)) is a continuous function with the property that f(q) = a^q  when a > 0 and q is a rational number, positive, negative or zero. So we define a^x = exp(x ln(a)) for all real numbers x. 

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2 .Real Numbers
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3. Derivative Motivation
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3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x)  (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
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3. Previous Rules Combined
3. d/dx for Polynomials
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3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
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3. Quotient Rule: tan & cot
3.  Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
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5. Indefinite Integrals A
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Advanced Topics

Limit Properties Algebraically
Pigeon Hole Principle
Bolzano
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
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From Lipschitz Continuity

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