Exercises - Graphs of Simple Functions, their Inverses, and Compositions

  1. Do the following for each:

    1. graph the function;
    2. label with coordinates the number and/or nature of points indicated;
    3. indicate the implicit domain and the corresponding image/range;
    4. find the inverse function, or say no such inverse exists as appropriate (be careful to state the domain of such inverses if they disagree with the implicit domain);
    5. and answer any additional questions provided.

    1. $f(x) = x+3$;   label $2$ points (the $x$ and $y$-intercepts)

    2. $f(x) = x-5$;   label $2$ points (the $x$ and $y$-intercepts)

    3. $f(x) = 2x$;   label $2$ points (one of which is the $y$ intercept); does this function represent a vertical dilation or contraction?

    4. $f(x) = \cfrac{x}{3}$;   label $2$ points (one of which is the $y$-intercept)

    5. $\displaystyle{f(x) = \left\{ \begin{array}{lll} -x & \textrm{ if } & x \gt 2\\ x & \textrm{ if } & x \le 2 \end{array} \right.}$;

      Noting $2$ points (both at $x=2$, drawing one "filled-in" to indicate it's part of the domain and the other drawn "open" to indicate it's not part of the domain but there are points on the graph of $f$ that are arbitrarily close to it.)

    6. $f(x)=x^3$;   label $3$ points ($x=-1,1, \textrm{ and } 2$); state the corresponding limit fact for $\lim_{x \rightarrow -\infty} f(x)$ and $\lim_{x \rightarrow \infty} f(x)$

    7. $f(x)=x^4$;   label $3$ points ($x=-1,1, \textrm{ and } 2$); state the corresponding limit fact for $\lim_{x \rightarrow -\infty} f(x)$ and $\lim_{x \rightarrow \infty} f(x)$

    8. $f(x)=x^5$;   label $3$ points ($x=-1,1, \textrm{ and } 2$); state the corresponding limit fact for $\lim_{x \rightarrow -\infty} f(x)$ and $\lim_{x \rightarrow \infty} f(x)$

    9. $f(x)=\sqrt[3]{x}$;   label $3$ points ($x=-1,1, \textrm{ and } 8$)

    10. $f(x)=\sqrt[4]{x}$;   label $2$ points ($x=1$ and $x=16$)

    11. $f(x)=\sqrt[5]{x}$;   label $3$ points ($x=-1,1, \textrm{ and } 32$)

    12. $f(x)=4^x$;   label $3$ points (including $x=-1,0, \textrm{ and } 1$); identify any horizontal asymptotes by writing the corresponding statement involving limits.

    13. $f(x)=(1/3)^x$;   label $3$ points (including $x=-1,0, \textrm{ and } 1$); identify any horizontal asymptotes by writing the corresponding statement involving limits.

    14. $f(x)=\log_3 x$   label $3$ points (including $x=\frac{1}{3}, 1, \textrm{ and } 3$)

    15. $f(x)=\log_{\frac{1}{4}} x$   label $3$ points (including $x=\frac{1}{4}, 1, \textrm{ and } 4$)

    click to see graphs (a)-(o)

  2. Graph the following after thinking about the simpler functions that can be composed together (and the order in which they are composed) to form them. Then state the implicit domain and image/range of each.

    1. $f(x) = 3x - 4$

    2. $f(x) = -x^2 + 2$

    3. $f(x) = -3|x| - 2$

    4. $f(x) = |x^2 - 4|$

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

    6. $f(x) = |-x^3|$

    7. $f(x) = 2^x - 1$

    8. $f(x) = |\log_{1/2} x|$

    click to see graphs (a)-(h)