Math 138 Review for Final

Linear Functions:

  1. You are about to take a trip for 7 days here are two rental offers: (i) Hertz will give you 28 cents per mile and $47 per day, and (ii) Avis will give you 27 cents per mile and $48 per day. Then sketch the cost functions for Hertz and Avis together.

Concept of Inverse Functions:

  1. The graph of the function $f$ is given below:
    final-prac__2.png

    1. Suppose we restrict the domain of $f$ to be $[0,4],$ and if $f(0)=0,$ $f(1)=1,$ and $f(4)=4.$ Find $f^{-1}(0),$ $f^{-1}(1)$ and $f^{-1}(4).$

    2. Suppose we restrict the domain of $f$ to be $[0,4],$ sketch $f^{-1}.$

Concepts of graphs of functions:

  1. Suppose the graph of $f$ is given below. Then
    final-prac__15.png

    1. graph $y=-f(x),$

    2. graph $y=f(-x),$

    3. graph MATH

    4. graph MATH

Polynomial Functions.

  1. A company's profit function $P(x)$ (when $x$ number of units are produced) satisfies the following conditions:

    1. the profit function breaks even at $x=100,300$ and $500$ units

    2. the profit function $P(x)$ changes signs (from $P(x)>0$ to $P(x)<0$ or vise versa) at $x=100$ but $P(x)$ does not change sign at $x=300$ and $500.$

    3. $P(50)<0.$

      Then (a) find a profit function which satisfies all the conditions mentioned above. (b) Predict if 400 units is produced, would the company be profitable?

  2. Sketch the graph which you have found in question above$.$

  3. For MATH,

    1. find the interval(s) where $f(x)>0,$

    2. sketch the graph for $f$.

Rational Functions.

  1. Find two rational functions $f$ satisfying ALL the following conditions:

    1. $f$ has two vertical asymptotes at $x=-1$ and $x=4$ respectively,

    2. $f$ has a horizontal asymptote at $y=-3,$

    3. $f(0)=1.$

  2. Suppose the average revenue per unit for producing $x$ number of units for a manufacturing company is given by
    MATH

    1. If $1,200$ units are produced, find the average revenue per unit.

    2. What is the maximum or minimum average revenue per unit?

Exponential and Logarithmic Functions.

  1. Find the inverse function for the followings and graph the function and its inverse together:

    1. $f(x)=3^{-x}$

    2. $f(x)=\log _{0.3}x$

  2. (New) If $f(x)=3^{x},$ find the inverse functions for $f(x+3)$ and $f(x)+3$ respectively.

  3. (New) If $f(x)=\log _{0.2}x,$ find the inverse functions for $f(x-3)$ and $f(x)-3$ respectively.

  4. (New) If $f(x)=-3^{x},$ find the inverse functions for $f(x+3)$ and $f(x)+3$ respectively.

  5. Solve the following equations. (Must show your works)

    1. $400e^{0.2x}=600$

    2. MATH.

  6. The number of some bacteria present after $t$ days of a laboratory experiment is given by MATH

    1. When will the bacteria population reach $3600?$

    2. Graph the function $p.$

  7. A model for the number of people $N$ in a college community who have heard a certain rumor is $N=P(1-e^{-0.15d}),$ where $P$ is the total population of the community and $d$ is the number of days that have elapsed since the rumor began.

    1. In a community of $9,100$ students, how many days will elapse before $4,000$ students have heard the rumor?

    2. Plot MATH step by step.

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