Exponential Functions

  1. Classify each of the following functions as a linear function, a quadratic function, a power function, an exponential function, or as “none of these”.
    \(a(x) = x - 5 + x^2\)
    \(b(x) = 405x\)
    \(c(x) = 3.14^x\)
    \(d(x) = 1.4142x^{2.71}\)
    \(f(x) = \sqrt{x}\)
    \(g(x) = x^2 - 2^x\)
    \(h(x) = 2^9 + x\)
    \(j(x) = 2.71^x\)
    \(k(x) = x^x\)
  2. Let \(p(x) = 3(1.4)^{-2x}.\) Calculate the following output values.
    \(p(3)\)
    \(p(-2)\)
    \(p(0)\)
    \(p(1/3)\)
  3. A tribble is an alien creature that, given an adequate food supply, is known to produce a litter of ten offspring every twelve hours. Due to this fact, if you start with a population of \(N_0\) tribbles, after \(t\) twelve-hour intervals the size of the population will be modelled by the function \(N(t) = N_0 (11)^t.\)
    1. In the formula for \(N(t)\) why is the base of the exponential eleven instead of ten?
    2. If you have an initial population of only two tribbles, how many tribbles will there be a day (24 hrs) later? What about a week later?
  4. The formula for the future value \(A\) of an initial investment of \(P\) dollars invested at an annual interest rate of \(r\) percent compounded annually for \(t\) years is given by the formula \[A= P(1+r)^t.\]
    1. If you deposit $420 in a savings account that gets \(3\%\) annual interest what will the account balance be in seven years? (Note that \(3\% = 0.03\,.\)) What would the account balance be in 70 years?
    2. A more realistic annual interest rate for a savings account at one of the Big Four US banks, like Chase of Bank of America, is \(0.01\%\). (Note that \(0.01\% = 0.0001\,.\)) If you deposit $420 into an account with this interest rate, what will the account balance be in seven years? What will the account balance be in 70 years?

  5. “Simplify” each of the following expressions. I.e., for each expression re-write it in a compact, nicer-looking form with fewer characters. Note that whether or not an expression can be “simplified” is entirely subjective; the purpose of this exercise is just to gain some fluency with the arithmetic of exponents.

    \(\left(10^0\right)^3 \)
    \(\left(10^3\right)^0 \)
    \(10^{\left(0^3\right)}\)
    \(10^{\left(3^0\right)}\)
    \(\left(10^1\right)^5 \)
    \(\left(10^5\right)^1\)
    \(10^{\left(1^5\right)}\)
    \(10^{\left(5^1\right)}\)
    \(\displaystyle 3^x\left(\frac{1}{2}\right)^x \)
    \(\displaystyle \frac{3^x}{\left(\frac{1}{2}\right)^x}\)
    \(\displaystyle 3^{-x}2^{2x}\)
    \(\displaystyle \sqrt{3^{6x}}\)
    \(\displaystyle \frac{\sqrt[3]{27^x}}{3^x}\)
    \(\displaystyle \frac{\sqrt{3x^2}}{x^2}\)
    \(\displaystyle \sqrt[3]{\frac{x^2}{3x^{-4}}}\)
    \(\displaystyle \left(\frac{2x^{-3}}{x^{11}}\right)^{-2}\)