Exponential functions

Exponential model of population growth

The natural exponential function

Logarithmic functions

Rules of log functions

  1. \(\ln\left(xy\right)=\ln x+\ln y\)
  2. \(\ln\frac{x}{y}=\ln\ x-\ln y\)
  3. \(\ln x^p=p\ln x\)
  4. \(\ln\ 1=0\)
  5. \(\ln e=1\)
  6. \(\ln e^x=x\)

Examples

  1. \(5e^{-3x}=16\)
  2. \(A\alpha e^{-\alpha x}=k\)
  3. \(e^x+4e^{-x}=4\)

Property of functions

Shifting graphs

Rules for shifting graphs of functions

  1. \(y=f\left(x\right)+c\): The graph for \(f\left(x\right)\) is moved up (down) by \(c\) if \(c>0\ \left(c<0\right)\)
  2. \(y=f(x+c)\): The graph is moved leftward (rightward) by \(c\) if \(c>0 (c<0)\)
  3. \(y=cf(x)\): The graph is stretched vertically if \(c>0\)
  4. \(y=f(-x)\): The graph is reflected about the \(y\) axis

Effect of a shift in supply

Tax credit vs. income deduction

New functions from old

Composite functions

Examples

  1. \(y=(x^3+x^2)^{50}\)
  2. \(y=e^{-(x-\mu)^2}\)

Symmetry of function

Inverse functions

Definition of inverse function

Examples

  1. \(y=4x-3\)
  2. \(y=\left(x+1\right)^{\frac{1}{5}}\)
  3. \(y=\frac{3x-1}{x+4}\)

Distance inn the plane: Circles

General functions