Math 151 Fall 04.

Test 1 (show all your works) Print Name MWF: 9 or 10 or 12.

  1. (30 points) Let $f(x)=e^{-x}.$

    1. Find the function $f_{1}$ so that the graph of $y=f_{1}(x)$ is being reflected along $x-axis.$ [hint: $f_{1}(x)=-e^{-x}]$

    2. Find the function $f_{2}$ so that the graph of $y=f_{2}(x)$ is being shifted to the right $2$ units and down $3$ units from $f_{1}.$ [hint: MATH

    3. Find the inverse function for $f_{2}(x)$ and graph the inverse function $y=f_{2}^{-1}(x)$ together with $y=f_{2}(x).$[hint: find the inverse using the followings:

      • the inverse for $y=e^{x}$ is $y=\ln x,$

      • the inverse for $y=e^{-x}$ is $y=-\ln x,$

      • the inverse for $y=-e^{-x}$ is $y=-\ln (-x),$

      • the inverse for $y=-e^{-(x-2)}$ is $y=-\ln (-x)+2$

      • the inverse for $y=-e^{-(x-2)}-3$ is $y=-\ln (-(x+3))+2.$

  2. (10 points) Find two rational functions $f$ which satisfy ALL the following conditions: [hint: try MATHand MATH

    1. the vertical asymptotes of $f$ are at $x=1$ and $x=-3$;

    2. the horizontal asymptote of $f$ is at $y=\frac{2}{3}$;

    3. $f(0)=3$.

  3. (10 points) If MATH,

    1. find the horizontal and vertical aysmptotes of $f$, [hint: h.a. is $y=200,$ and v.a. are $x=100$ and $x=200].$

    2. solve $f(x)>0.$ [hint: graph MATH first and solve $f(x)>0]$

  4. (20 points) Find a function whose graph (shape) could resemble the following. Note the $x-intercepts$ are at $x=-2,0$ and $2.$ [hint: try MATH
    graphics/test1-answers__46.png

  5. (10 points) If the graph of a function $f$ is given in the first quadrant. Then label the remaining functions $-f(-x)$,$-f(-x)-1$,$f(-(x+2))+3,$ whose graphs are given below. MATH

  6. (15 points) If MATH

    1. find the vertical asymptote of $f,$[hint: since $y=\log _{0.3}x$ has the v.a. $x=0,$and MATH is being shifted to the right $10,$so the v.a. for $y=f(x)$ is $x=10.]$

    2. find $f^{-1}$ (the inverse function of $f),$ [hint: the inverse for $y=\log _{0.3}x$ is $y=0.3^{x},$so MATH

    3. find $f^{-1}(0).$ [hint: $11].$

  7. (15 points) Let MATH Then

    1. find the vertex of this parabola, [hint: $(-40,-30)].$

    2. sketch the graph of $y=f(x),$

    3. find the domain and range for $f.$[hint: domain is $(-\infty ,\infty )$ and the range is $(-\infty ,-30].$]