Review Exercises (Set B)

  1. Find the distance between points $(1,f(1))$ and $(3,f(3))$ if $f(x)$ is defined in the following way: $$f(x) = \left\{ \begin{array}{ccl} 3x &,& x \lt 0\\ x+1 &,& 0 \le x \le 2\\ (x-2)^2 &,& x \gt 2 \end{array} \right.$$

  2. Given points $A = (5,-2)$ and $B = (2,3)$, find the following:

    1. The midpoint of the segment with $A$ and $B$ as endpoints
    2. An equation in point-slope form for the line that goes through $A$ and $B$
    3. The slope of any line perpendicular to the line that goes through $A$ and $B$
  3. Given $f(x) = \sqrt{x-5}$ and $g(x) = 2x^2 + x$, answer the following questions:

    1. Find the domain of $(f/g)(x)$

    2. Find and simplify $(f \circ g)$

    3. Find and simplify $\displaystyle{\frac{g(x+h) - g(x)}{h}}$, assuming $h \ne 0$

  4. Solve for $x$:

    1. $\displaystyle{x^{5/3} + 7x^{4/3} + 12x = 0}$

    2. $\displaystyle{\sqrt{x+9} - 3 = \sqrt{\displaystyle{\frac{x}{4}}}}$

    3. $\displaystyle{\frac{12}{x^2 - 1} = \frac{2x-5}{z-1} + \frac{2x+5}{x+1}}$

    4. $\displaystyle{|2x - 5| > 5}$

    5. $\displaystyle{x^2-11x+30 < 0}$

  5. Given $\displaystyle{f(x) = \frac{5x^3-1}{2-x^3}}$, find $f(x)$ and show $(f^{-1} \circ f)(x) = x$

  6. Graph $\displaystyle{f(x) = 2x^2 - 12x - 9}$ and identify the coordinates of the vertex and all intercepts

  7. List the transformations that can be applied to $y=|x|$, in the order they should be applied, to arrive at the graph of $\displaystyle{g(x) = -\frac{1}{3}|x-2| + 5}$. Then graph $y=g(x)$

  8. Graph $\displaystyle{f(x) = \left\{ \begin{array}{ccl} \sqrt{16-x^2} &,& x < 0\\ 2 &,& 0 \le x \le 2\\ 3x-1&,& x > 2 \end{array} \right.}$