5F13Final

 Question 1  Please circle either true or false,
    a. (True/False)In a geometric sequence, the common ratio is always positive.
    b. (True/False) A linear system of equations always has a solution.
    c. (True/False) Every function has an inverse.
    d. (True/False) Trigonometric equations do not always have unique solutions.
    e. (True/False) The domain of $f(x)=\tan ^{-1}(x)$ is all real numbers.
    f. (True/False) The function $\log _{a}(x)$ is defined for all real numbers.

2.Find the domain of the following function. Your answer should be in interval notation $f(x)={\frac {1}{\sqrt {x^{2}-x-2}}}$

3. Find f $\circ$ g and its domain if $f(x)=x^{2}+1\qquad g(x)={\sqrt {x-1}}$

4. Find the inverse of the following function $f(x)={\frac {3x}{2x-1}}$

5. Solve the following inequality. Your answer should be in interval notation. ${\frac {3x+5}{x+2}}\geq 2$

6. Factor the following polynomial completely,     $p(x)=x^{4}+x^{3}+2x-4$

7. Solve the following equation,      $2\log _{5}(x)=3\log _{5}(4)$

8. Solve the following equation,      $3^{2x}+3^{x}-2=0$

9. Solve the following system of equations

${\begin{aligned}2x+3y&=&1\\-x+y&=&-3\end{aligned}}$

10. Write the partial fraction decomposition of the following,

${\frac {x+2}{x^{3}-2x^{2}+x}}$

11. Solve the following equation in the interval $[0,2\pi )$

$\sin ^{2}(\theta )-\cos ^{2}(\theta )=1+\cos(\theta )$

12. Given that $\sec(\theta )=-2$ and $\tan(\theta )>0$, find the exact values of the remaining trig functions.

13. Give the exact value of the following if its defined, otherwise, write undefined.
$(a)\sin ^{-1}(2)\qquad \qquad (b)\sin \left({\frac {-32\pi }{3}}\right)\qquad \qquad (c)\sec \left({\frac {-17\pi }{6}}\right)$

14. Prove the following identity,

${\frac {1-\sin(\theta )}{\cos(\theta )}}={\frac {\cos(\theta )}{1+\sin(\theta )}}$

15. Find an equivalent algebraic expression for the following,

$\cos(\tan ^{-1}(x))$

16. Graph the following,

$-x^{2}+4y^{2}-2x-16y+11=0$

17. Graph the following function,

$f(x)=\log _{2}(x+1)+2$

Make sure to label any asymptotes, and at least two points on the graph.

18. Graph the following function,

$f(x)=\left({\frac {1}{3}}\right)^{x+1}+1$

Make sure to label any asymptotes, and at least two points on the graph.

19. Consider the following function,

$f(x)=-\sin \left(3x+{\frac {\pi }{2}}\right)+1$

     a. What is the amplitude?
     b. What is the period?
     c. What is the phase shift?
     d. What is the vertical shift?
     e. Graph one cycle of f(x). Make sure to label five key points.

20. Consider the following rational function,

$f(x)={\frac {x^{2}+x-2}{x^{2}-1}}$

     a. What is the domain of f?
     b. What are the x and y-intercepts of f?
     c. What are teh vertical and horizontal asymptotes of f, if any? Does f have any holes?
     d. Graph f(x). Make sure to include the information you found above.

21. Find the sum

$5+9+13+\cdots +49$

22. Consider the following sequence,

$-3,1,-{\frac {1}{3}},{\frac {1}{9}},-{\frac {1}{27}},\cdots$

     a. Determine a formula for $a_{n}$, the n-th term of the sequence.
     b. Find the sum $\displaystyle {\sum _{k=1}^{\infty }a_{k}}$