Problems

1. (a). Solve for $u$.

   $\dfrac{u+5}{u+7}=\dfrac{u-6}{u-3}-1$

   (a). Solve for $y$.

   $\dfrac{y-3}{y-2}+1=\dfrac{y+5}{y+2}$

2. (a). Solve. $(v-2)^2-40=0$, where $v$ is a real number.

    Simplify your answer as much as possible.

   (b). Solve. $(w+6)^2-64=0$, where $w$ is a real number.

    Simplify your answer as much as possible.

3. Solve the quadratic equation by completing the square.

   (a). $x^2-8x+1$

   (a). $x^2+6x-5$

4. Use the quadratic formula to solve for $x$. Round your answers to the nearest hundredth.

   (a). $4x^2+5x-2=0$

   (b). $2x^2=9x-5$

5. Graph the parabola.

   (a). $y=(x+2)^{2}-3$

   (b). $y=(x-2)^{2}+1$

6. Graph the parabola.

   (a). $y=2x^2+8x+10$

   (b). $y=-x^2+4x-2$

7. (a). A ball is thrown vertically upward. After $t$ seconds, its height $h$ (in feet) is given by the function $h(t)=104t-16t^2$. What is the maximum height that the ball will reach? Do not round your answer.

   (b). A vehicle factory manufactures cars. The unit cost (the cost in dollars to make each car) depends on the number of cars made. If $x$ cars are made, then the unit cost is given by the function $C(x)=0.8x^2-432x+68031$. What is the minimum unit cost? Do not round your answer.

   (c). An aircraft factory manufactures airplane engines. The unit cost $C$ (the cost in dollars to make each airplane engine) depends on the number of engines made. If $x$ engines are made, then the unit cost is given by the function $C(x)=0.3x^2-120x+23,126$. How many engines must be made to minimize the unit cost? Do not round your answer.

8. Graph the solution to the following inequality on the number line.

   (a). $x^2\le -2x$

   (b). $x^2>-6x$

9. Solve the following inequalities. Write your answer as an interval or union of intervals.

   (a). $\dfrac{-12}{1-x}\gt x-2$

   (b). $\dfrac{1}{x+5}\le \dfrac{2}{x+13}$

10. (a). Suppose that the functions $r$ and $s$ are defined for all real numbers $x$ as follows.

    $r(x)=4x-2$

    $s(x)=x-6$

    Write the expressions for $(r-s)(x)$ and $(r+s)(x)$ and evaluate $(r\cdot s)(4)$.

    (b). Suppose that the functions $f$ and $g$ are defined for all real numbers $x$ as follows.

    $f(x)=2x^2$

    $g(x)=x-5$

    Write the expressions for $(g\cdot f)(x)$ and $(g+f)(x)$ and evaluate $(g-f)(-2)$.

11. (a). Suppose that the functions $h$ and $g$ are defined as follows.

     $h(x)=x+3$

     $g(x)=(x+6)(x-6)$

     (i). Find $\left (\dfrac{h}{g}\right)(-1).$

     (ii). Find all values that are NOT in the domain of $\dfrac{h}{g}$.

    (b). Suppose that the functions $h$ and $g$ are defined as follows.

     $g(x)=(3+x)(-5+x)$

     $f(x)=-2-x$

     (i). Find $\left (\dfrac{g}{f}\right)(5).$

     (ii). Find all values that are NOT in the domain of $\dfrac{g}{f}$.

12. (a). Suppose that the functions $q$ and $r$ are defined as follows.

     $q(x)=x^2+7$

     $r(x)=\sqrt{x+4}$

    Find $(q\circ r)(5)$ and $(r\circ q)(5)$.

    (b). Suppose that the functions $u$ and $w$ are defined as follows.

     $u(x)=4x-1$

     $w(x)=-3x-3$

    Find $(w\circ u)(-4)$ and $(u\circ w)(-4)$.

13. (a). The one-to-one function $g$ is defined below.

     $g(x)=\dfrac{7x}{9x-8}$

     Find $g^{-1}(x)$.

     Also state the domain and range of $g^{-1}$ in interval notation.

    (b). The one-to-one function $h$ is defined below.

     $h(x)=\dfrac{x-7}{6x+1}$

     Find $h^{-1}(x)$.

     Also state the domain and range of $h^{-1}$ in interval notation.

14. (a). Rewrite as a logarithmic equation.

     $3^4=81$

    (b). Rewrite as an exponential equation.

     $\log_8 64=2$

15. (a). Graph the logarithmic function.

     $g(x)=4\log_{1/3}x$

    (b). Graph the logarithmic function.

     $g(x)=3+\log_{1/2}x$

16. (a). Use the properties of logarithms to expand

     $\log\left [\dfrac{yz^3(y+3)^5}{x^2}\right ]$

    (b). Use the properties of logarithms to expand

      $\log{\sqrt{\dfrac{y}{(z+2)^9x^4}}}$

17. (a). Write the expression as a single logarithm.

     $\dfrac{1}{3}\log_m w+2\log_m x-\log_m y$

    (b). Write the expression as a single logarithm.

     $2\log_a y+2\left (\log_a z-3\log_a w\right )$

18. Use the change of base formula to compute. Round your answer to the nearest thousandth.

      (a). $\log_7 6$

     (b). $\log_9 \dfrac{1}{5}$

19. (a). Solve for $x$.

     $\log_2(5-4x) = 3$

    (b). Solve for $x$.

     $2+\log_3(x-7) = 5$

20. (a). Solve for $x$.

     $\log_6 2 +\log_6 (x-7)=\log_6 10$

    (b). Solve for $x$.

      $\log 5 = \log(x+1)-\log16$

    (c). Solve for $x$.

     $\log_9 x-\log_9(x+4) = \log_9 2$

21. (a). Solve for $x$.

     $125=25^{-x+3}$

    (b). Solve for $x$.

     $2^{-8x}=16^{5-x}$

22. (a). Solve for $x$.

     $9^{x-8}=13^{7x}$

    (b). Solve for $x$.

     $2^{10x}=7^{-x-9}$

23. (a). A desktop computer is purchased for $\$3000$. Each year, its value is $70\%$ of its value the year before. After how many years will the laptop computer be worth $\$900$ or less?

     Write the smallest possible whole number answer.

    (b). A principal of $\$2700$ is invested at $8.25\%$ interest, compounded annually. How many years will it take to accumulate $\$7000$ or more in the account?

     Write the smallest possible whole number answer.