Exercises - Linear Functions and Mobius Transformations

  1. Find the linear function $f(x)$ whose graph has the given characteristics.

    1. $m = \frac{2}{9}$, $y$-intercept $(0,4)$

    2. $m = -\frac{8}{3}$, $y$-intercept $(0,-2)$

    3. $m = -5$, $y$-intercept $(0,-\frac{2}{3})$

    4. $m = -2$, passes through $(-5,1)$

    5. $m = \frac{2}{3}$, passes through $(-4,-5)$

    6. passes through $(-3,7)$ and $(-1,-5)$

    7. $f(-5) = -3$ and $f(5) = 1$

    1. $f(x) = \frac{2}{9} x + 4$
    2. $f(x) = -\frac{8}{3} x - 2$
    3. $f(x) = -5x-\frac{2}{3}$
    4. $f(x) = -2x -9$
    5. $f(x) = \frac{2}{3}x - \frac{7}{3}$
    6. $f(x) = -6x-11$
    7. $f(x) = \frac{2}{5} x - 1$
  2. Determine if the following relations correspond to linear functions whose graphs are parallel or perpendicular (specify which), or something else.

    1. $y = 3x + 1$ and $2y = 6x -7$

    2. $y + 3x = 1$ and $y = \frac{1}{3} x + 1$

    3. $2x + 5y = 4$ and $x = -\frac{5}{2} y - 7$

    4. $y = 2x - 1$ and $y = -\frac{1}{2} x + 3$

    5. $y = 7-x$ and $y = x+3$

    6. $y + 3x = 2y - x$ and $y = 4x + 1$

    1. parallel
    2. perpendicular
    3. parallel
    4. perpendicular
    5. perpendicular
    6. parallel

  3. Find the inverse of each function given, if it exists.

    1. $f(x) = -\frac{2}{3} x + 4$

    2. $f(x) = 6 x - \frac{1}{2}$

    3. $f(x) = 7$

    4. $f(x) = \cfrac{x+4}{x-3}$

    5. $f(x) = \cfrac{5x-3}{2x+1}$

    6. $f(x) = \cfrac{x+6}{3x-4}$

    1. $f^{-1}(x) = -\frac{3}{2} x + 6$

    2. $f^{-1}(x) = \frac{1}{6} x + \frac{1}{12}$

    3. no inverse exists -- $f$ graphs as a horizontal line which spectacularly fails the horizontal line test

    4. $f^{-1}(x) = \cfrac{3x+4}{x-1}$

    5. $f^{-1}(x) = \cfrac{-x-3}{2x-5}$

    6. $f^{-1}(x) = \cfrac{4x+6}{3x-1}$

  4. Find the inverse of $f(x) = \cfrac{6x + 8}{2x+1}$ in two different ways.

    For one of the ways, do the division first, following this with a "socks-and-shoes" argument.