Final Review for Math 121, Summer 2012

Topics we covered.

1. composite functions:
   1. the weird notation $f\circ g$, what it means
   2. computing $(f\circ g)(x)$ when $f(x)$ and $g(x)$ are given
   3. computing the domain of $f\circ g$
2. one-to-one functions, inverses
   1. the definition of one-to-one
   2. how to tell if a function is one-to-one: the HLT
   3. the definition and idea of the inverse function $f^{-1}$
   4. how to sketch the graph of $f^{-1}$ given the graph of $f$
   5. how to find the formula for $f^{-1}(x)$ given the formula for $f(x)$
3. exponential functions
   1. basic idea (constant multiplicative rate of growth) and formulæ
   2. general shape of graph, including intercepts and asymptotes
   3. the special base, Euler's number $e$
   4. simple manipulations using the power laws and the fact that $a^x$ is 1-1
4. logarithms
   1. definition
   2. basic computations
   3. the domain
   4. the graph, including intercepts and asymptotes
   5. the Log Laws
      1. logs of products
      2. logs of quotients
      3. logs of powers
      4. chage of base
5. solving exponential and logarithmic equations
   1. use log laws or laws of exponents to combine terms
   2. get the variable out of the exponent or the log by taking the log or exponent
   3. the only other type of equation which might occur in this class is one where you use laws of exponents and a substitution to turn it into a quadratic equation. If so, don't forget to check your answers at the end, some of them may not work in the substitution
6. applications of logs and exponentials
   1. exponential (or "uninhibited") growth -- e.g., of bacterial or insect populations
   2. radioactive decay
   3. Newton's Law of Cooling
   4. in just about all applications, you can assume the underlying function in the model should have the form $f(t)=Ce^{kt}$, where $C$ is the starting population/quantiy/temperature difference, and $k$, which usually has to be solved for, is associated to the rate of the process.