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College Algebra -- Fall 2013 -- final exam

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12 problems, 10 points each

  • Provide as much explanation for every step as you can.
  • Unless requested, no decimal representation of the answers is necessary.
  • Start every problem at the top of the page.
  1. Solve the following equation: $\sqrt{x^2-7}-3=0$.
  2. Give an example of an even function, an odd function, and a function that's neither. Provide formulas.
  3. Plot the graph of the function $y=f(x)$, where $x$ is time in hours and $y=f(x)$ is the parking fee over $x$ hours, which is computed as follows: free for the first hour, $\$1$ per every full hour for the next $3$ hours, flat $\$5$ for anything longer.
  4. For the polynomial $f(x)=2(x-2)^2(x+3/7)^3$, find its $x$-intercepts and its large scale behavior, i.e., $\lim _{x\to \pm \infty}f(x)$.
  5. Represent the composition of these two functions: $f(x)=\ln x,g(u)=u^2-3u$, as a single function of variable $x$.
  6. Represent this function: $f(x)=e^{\sqrt{x-1}}$, as the composition of two functions of variables $x$ and $u$.
  7. Function $y=f(x)$ is given below by a list of some of its values. Is the function one-to one? What about its inverse? $$\begin{array}{lllll}x&: &0 &1 &2 &3 &4 \\y=f(x)&: &0 &1 &2 &1 &2 \end{array}$$
  8. Plot the graph of the function $f(x)=\frac{1}{x-1}$ and the graph of its inverse. Identify its important features.
  9. To what power should you raise $3$ to get $10$?
  10. Find the domain, the range, and the asymptotes of the function $f(x)=\ln (x-3)+\ln 3$.
  11. A city loses 5% of its population every year. How long will it take to lose 30%?
  12. Set up a system of linear equations -- but do not solve -- for the following problem: "A mix of coffee is to be prepared from: Kenyan coffee - $\$3$ per pound and Colombian coffee - $\$5$ per pound. How much of each do you need to have $10$ pounds of blend with $\$3.50$ per pound?"