Classification

Elementary Functions

• An elementary function is a function which belongs to the class of functions consisting of the polynomials, the exponential functions, the logarithmic functions, the trigonometric functions, the inverse trigonometric functions, and the functions obtained from those listed by the four arithmetic operations and/or by composition, applied finitely many times (see d2.3-infi2)
  • All elementary functions are continuous on their domains.

Algebraic functions

Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.

• Polynomials
  • Limit
• Rational functions: A ratio of two polynomials.
• nth root

Transcendental functions

Transcendental functions are functions that are not algebraic.

Exponential Functions

• $f(x)=a^x$ where $a>0$

  • $\text{dom}(f)=\mathbb{R}$
  • $f$ is continuous on $\mathbb{R}$
  • one-to-one function
  • x-intercept: none
  • y-intercept: $(0,1)$
  • $\text{Im}f=(0,\infty )$
  • Inverse Function $f^{-1}={\log }_a(x)$ (where $a\ne 1$)
  • limit
    • (Increasing) $a>1⟹$
      • $\underset{x\to -\infty }{\lim }f(x)=0$
      • $\underset{x\to \infty }{\lim }f(x)=\infty$
    • (Decreasing) $0<a<1⟹$
      • $\underset{x\to -\infty }{\lim }f(x)=\infty$
      • $\underset{x\to \infty }{\lim }f(x)=0$
  • Derivative

• (natural exponential function) $f(x)=e^x$

• generalizations of exponential function:

  • $f(x)=a{b}^{cx+d}$, for $a,b,c,d\in \mathbb{R}$ and $b>0$
    • $f(x)$ is an exponentially increasing function if $c>0$, and is an exponentially decreasing function if $c<0$.
  • $f(x)=a{b}^x=a(1+r{)}^x$, where:
    • $a$ is the initial amount
    • $b$ is the base
      • if $0<b<1$, then $f$ is an exponential decay
    • $r$ is the growth rate (if $r>0$) or the decay rate (if $r<0$) per time period
    • $x=0,1,2,\dots$ is the number of time periods that have passed
  • (exponential decay) $f(x):[0,\infty ]\to \mathbb{R}$, if $\exists \lambda >0:\frac{df}{dx}=-\lambda f$ (or equivalently: $\exists \lambda >0,a>0:f(x)=a{e}^{-\lambda x}$), where:
    • $a=f(0)$ is the initial amount
    • $\lambda =-\ln b$ is the decay constant (where $b$ is the base)
    • half-life: $t_{1/2}=\frac{\ln (2)}{\lambda }$
    • mean lifetime: $\tau =\frac{1}{\lambda }$

Logarithmic Functions

• $f(x)={\log }_a(x)$ where $a>0$ and $a\ne 1$
  • $\text{dom}f=(0,\infty )$
  • $\text{Im}f=\mathbb{R}$
  • $f$ is continuous on $(0,\infty )$
  • one-to-one function
  • X-intercept: $(1,0)$
  • Y-intercept: none
  • Asymptotes
    • Vertical asymptote at $x=0$
    • Horizontal asymptote as $x$ approaches infinity
  • Inverse Function $f^{-1}=a^x$
  • limit
    • (Increasing) $a>1⟹$
      • $\underset{x\to 0^{+}}{\lim }{\log }_a(x)=-\infty$
      • $\underset{x\to \infty }{\lim }{\log }_a(x)=\infty$
    • (Decreasing) $0<a<1⟹$
      • $\underset{x\to 0^{+}}{\lim }{\log }_a(x)=\infty$
      • $\underset{x\to \infty }{\lim }{\log }_a(x)=-\infty$
  • Derivative

Power functions

• $f(x)=a{x}^r$
• $a,r\in \mathbb{R}$

Periodic functions

Trigonometric Functions

• Limit
• Derivative

Inverse trigonometric functions

• Limit
• Derivative