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Mathematics LibreTexts

1.1E: Exercises for Section 1.1

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    69574
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    For exercises 1 - 6, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

    1)

    \(x\) \(y\) \(x\) \(y\)
    -3 9 1 1
    -2 4 2 4
    -1 1 3 9
    0 0    
    Answer:

    a. Domain = {\(−3,−2,−1,0,1,2,3\)},  Range = {\(0,1,4,9\)}

    b. Yes, a function

    2)

    \(x\) \(y\) \(x\) \(y\)
    -3 -2 1 1
    -2 -8 2 8
    -1 -1 3 -2
    0 0    

    3)

    \(x\) \(y\) \(x\) \(y\)
    1 -3 1 1
    2 -2 2 2
    3 -1 3 3
    0 0    
    Answer:

    a. Domain = {\(0,1,2,3\)}, Range = {\(−3,−2,−1,0,1,2,3\)}

    b. No, not a function

    4)

    \(x\) \(y\) \(x\) \(y\)
    1 1 5 1
    2 1 6 1
    3 1 7 1
    4 1    

    5)

    \(x\) \(y\) \(x\) \(y\)
    3 3 15 1
    5 2 21 2
    8 1 33 3
    10 0    
    Answer:

    a. Domain = {\(3,5,8,10,15,21,33\)}, Range = {\(0,1,2,3\)}

    b. Yes, a function

    6)

    \(x\) \(y\) \(x\) \(y\)
    -7 11 1 -2
    -2 5 3 4
    -2 1 6 11
    0 -1    

     

    For exercises 7 - 13, find the values for each function, if they exist, then simplify.

    a. \(f(0)\)     b. \(f(1)\)      c. \(f(3)\)      d. \(f(−x)\)      e. \(f(a)\)      f. \(f(a+h)\)

    7) \(f(x)=5x−2\)

    Answer:
    a. \(−2\)      b. \(3\)      c. \(13\)      d. \(−5x−2\)      e. \(5a−2\)      f. \(5a+5h−2\)

    8) \(f(x)=4x^2−3x+1\)

    9) \(f(x)=\dfrac{2}{x}\)

    Answer:
    a. Undefined      b. \(2\)      c. \(\frac{2}{3}\)      d. \(−\dfrac{2}{x}\)      e. \(\dfrac{2}{a}\)      f. \(\dfrac{2}{a+h}\)

    10) \(f(x)=|x−7|+8\)

    11) \(f(x)=\sqrt{6x+5}\)

    Answer:
    a. \(\sqrt{5}\)      b. \(\sqrt{11}\)      c. \(\sqrt{23}\)      d. \(\sqrt{−6x+5}\)      e. \(\sqrt{6a+5}\)      f. \(\sqrt{6a+6h+5}\)

    12) \(f(x)=\dfrac{x−2}{3x+7}\)

    13) \(f(x)=9\)

    Answer:
    a. 9      b. 9      c. 9      d. 9      e. 9      f. 9

     

    For exercises 14 - 21, find the domain, range, and all zeros/intercepts, if any, of the functions.

    14) \(f(x)=\dfrac{x}{x^2−16}\)

    15) \(g(x)=\sqrt{8x−1}\)

    Answer:
    \(x≥\frac{1}{8};\quad y≥0;\quad x=\frac{1}{8}\);   no y-intercept

    16) \(h(x)=\dfrac{3}{x^2+4}\)

    17) \(f(x)=−1+\sqrt{x+2}\)

    Answer:
    \(x≥−2;\quad y≥−1;\quad x=−1;\quad y=−1+\sqrt{2}\)

    18) \(f(x)=1x−\sqrt{9}\)

    19) \(g(x)=\dfrac{3}{x−4}\)

    Answer:
    \(x≠4;\quad y≠0\);   no x-intercept;   \(y=−\frac{3}{4}\)

    20) \(f(x)=4|x+5|\)

    21) \(g(x)=\sqrt{\dfrac{7}{x−5}}\)

    Answer:
    \(x>5;\quad y>0\);   no intercepts

     

    For exercises 22 - 27, set up a table to sketch the graph of each function using the following values: \(x=−3,−2,−1,0,1,2,3.\)

    22) \(f(x)=x^2+1\)

    \(x\) \(y\) \(x\) \(y\)
    -3 10 1 2
    -2 5 2 5
    -1 2 3 10
    0 1    

    23) \(f(x)=3x−6\)

    \(x\) \(y\) \(x\) \(y\)
    -3 -15 1 -3
    -2 -12 2 0
    -1 -9 3 3
    0 -6    
    Answer:
    An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -3 to 3. The graph is of the function “f(x) = 3x - 6”, which is an increasing straight line. The function has an x intercept at (2, 0) and the y intercept is not shown.

    24) \(f(x)=\frac{1}{2}x+1\)

    \(x\) \(y\) \(x\) \(y\)
    -3 \(-\frac{1}{2}\) 1 \(\frac{3}{2}\)
    -2 0 2 2
    -1 \(\frac{1}{2}\) 3 \(\frac{5}{2}\)
    0 1    

    25) \(f(x)=2|x|\)

    \(x\) \(y\) \(x\) \(y\)
    -3 6 1 2
    -2 4 2 4
    -1 2 3 6
    0 0    
    Answer:
    An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -2 to 6. The graph is of the function “f(x) = 2 times the absolute value of x”. The function decreases in a straight line until it hits the origin, then begins to increase in a straight line. The function x intercept and y intercept are at the origin.

    26) \(f(x)=-x^2\)

    \(x\) \(y\) \(x\) \(y\)
    -3 -9 1 -1
    -2 -4 2 -4
    -1 -1 3 -9
    0 0    

    27) \(f(x)=x^3\)

    \(x\) \(y\) \(x\) \(y\)
    -3 -27 1 1
    -2 -8 2 8
    -1 -1 3 27
    0 0    
    Answer:
    An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -27 to 27. The graph is of the function “f(x) = x cubed”. The curved function increases until it hits the origin, where it levels out and then becomes even. After the origin the graph begins to increase again. The x intercept and y intercept are both at the origin.

     

    For exercises 28 - 35, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph:

    a. Domain and range

    b. \(x\) -intercept, if any (estimate where necessary)

    c. \(y\)-Intercept, if any (estimate where necessary)

    d. The intervals for which the function is increasing

    e. The intervals for which the function is decreasing

    f. The intervals for which the function is constant

    g. Symmetry about any axis and/or the origin

    h. Whether the function is even, odd, or neither

    28)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is circle, with x intercepts at (-1, 0) and (1, 0) and y intercepts at (0, 1) and (0, -1).

    29)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is curved. The relation decreases until it hits the point (-1, 0), then increases until it hits the point (0, 1), then decreases until it hits the point (1, 0), then increases again.

    Answer:
    Function;
    a. Domain: all real numbers, range: \(y≥0\)
    b. \(x=±1\)
    c. \(y=1\)
    d. \(−1<x<0\) and \(1<x<∞\)
    e. \(−∞<x<−1\) and \(0<x<1\)
    f. Not constant
    g. \(y\)-axis
    h. Even

    30)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a parabola. The curved relation increases until it hits the point (2, 3), then begins to decrease. The approximate x intercepts are at (0.3, 0) and (3.7, 0) and the y intercept is is (-1, 0).

    31)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is curved. The curved relation increases the entire time. The x intercept and y intercept are both at the origin.

    Answer:
    Function;
    a. Domain: all real numbers, range: \(−1.5≤y≤1.5\)
    b. \(x=0\)
    c. \(y=0\)
    d. all real numbers
    e. None
    f. Not constant
    g. Origin
    h. Odd

    32)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a sideways parabola, opening up to the right. The x intercept and y intercept are both at the origin and the relation has no points to the left of the y axis. The relation includes the points (1, -1) and (1, 1)

    33)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a horizontal line until the point (-2, -2), then it begins increasing in a straight line until the point (2, 2). After these points, the relation becomes a horizontal line again. The x intercept and y intercept are both at the origin.

    Answer:
    Function;
    a. Domain: \(−∞<x<∞\), range: \(−2≤y≤2\)
    b. \(x=0\)
    c. \(y=0\)
    d. \(−2<x<2\)
    e. Not decreasing
    f. \(−∞<x<−2\) and \(2<x<∞\)
    g. Origin
    h. Odd

    34)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that is a horizontal line until the origin, then it begins increasing in a straight line. The x intercept and y intercept are both at the origin and there are no points below the x axis.

    35)

    An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a relation that starts at the point (-4, 4) and is a horizontal line until the point (0, 4), then it begins decreasing in a curved line until it hits the point (4, -4), where the graph ends. The x intercept is approximately at the point (1.2, 0) and y intercept is at the point (0, 4).

    Answer:
    Function;
    a. Domain: \(−4≤x≤4\), range: \(−4≤y≤4\)
    b. \(x=1.2\)
    c. \(y=4\)
    d. Not increasing
    e. \(0<x<4\)
    f. \(−4<x<0\)
    g. No Symmetry
    h. Neither

     

    For exercises 36 - 41, for each pair of functions, find a. \(f+g\) b. \(f−g\) c. \(f⋅g\) d. \(f/g\). Determine the domain of each of these new functions.

    36) \(f(x)=3x+4,\quad g(x)=x−2\)

    37) \(f(x)=x−8,\quad g(x)=5x^2\)

    Answer:
    a. \(5x^2+x−8\); all real numbers
    b. \(−5x^2+x−8\); all real numbers
    c. \(5x^3−40x^2\); all real numbers
    d. \(\dfrac{x−8}{5x^2};\quad x≠0\)

    38) \(f(x)=3x^2+4x+1,\quad g(x)=x+1\)

    39) \(f(x)=9−x^2,\quad g(x)=x^2−2x−3\)

    Answer:
    a. \(−2x+6\); all real numbers
    b. \(−2x^2+2x+12\); all real numbers
    c. \(−x^4+2x^3+12x^2−18x−27\); all real numbers
    d. \(−\dfrac{x+3}{x+1};\quad x≠−1,3\)

    40) \(f(x)=\sqrt{x},\quad g(x)=x−2\)

    41) \(f(x)=6+\dfrac{1}{x},\quad g(x)=\dfrac{1}{x}\)

    Answer:

    a.  \(6+\dfrac{2}{x};\quad x≠0\)

    b.  \(6; \quad x≠0\)

    c.  \(6x+\dfrac{1}{x^2};\quad x≠0\)

    d.  \(6x+1;\quad x≠0\)

     

    For exercises 42 - 48, for each pair of functions, find a. \((f∘g)(x)\) and b. \((g∘f)(x)\) Simplify the results. Find the domain of each of the results.

    42) \(f(x)=3x,\quad g(x)=x+5\)

    43) \(f(x)=x+4,\quad g(x)=4x−1\)

    Answer:
    a. \(4x+3\); all real numbers
    b. \(4x+15\); all real numbers

    44) \(f(x)=2x+4,\quad g(x)=x^2−2\)

    45) \(f(x)=x^2+7,\quad g(x)=x^2−3\)

    Answer:
    a. \(x^4−6x^2+16\); all real numbers
    b. \(x^4+14x^2+46\); all real numbers

    46) \(f(x)=\sqrt{x}, \quad g(x)=x+9\)

    47) \(f(x)=\dfrac{3}{2x+1},\quad g(x)=\dfrac{2}{x}\)

    Answer:

    a. \(\dfrac{3x}{4+x};\quad x≠0,−4\)

    b. \(\dfrac{4x+2}{3};\quad x≠−\frac{1}{2}\)

    48) \(f(x)=|x+1|,\quad g(x)=x^2+x−4\)

     

    49) The table below lists the NBA championship winners for the years 2001 to 2012.

    Year Winner
    2001 LA Lakers
    2002 LA Lakers
    2003 Sam Antonio Spurs
    2004 Detroit Pistons
    2005 Sam Antonio Spurs
    2006 Miami Heat
    2007 Sam Antonio Spurs
    2008 Boston Celtics
    2009 LA Lakers
    2010 LA Lakers
    2011 Dallas Mavericks
    2012 Miami Heat

    a.  Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not.

    b.  Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not.

    Answer:
    a. Yes, because there is only one winner for each year.
    b. No, because there are three teams that won more than once during the years 2001 to 2012.

    50) [T] The area \(A\) of a square depends on the length of the side s.

    a.  Write a function \(A(s)\) for the area of a square.

    b.  Find and interpret \(A(6.5)\).

    c.  Find the exact and the two-significant-digit approximation to the length of the sides of a square with area 56 square units.

    51) [T] The volume of a cube depends on the length of the sides \(s.\)

    a.  Write a function \(V(s)\) for the area of a square.

    b.  Find and interpret \(V(11.8)\).

    Answer:
    a. \(V(s)=s^3\)
    b. \(V(11.8)≈1643\); a cube of side length 11.8 each has a volume of approximately 1643 cubic units.

    52) [T] A rental car company rents cars for a flat fee of $20 and an hourly charge of $10.25. Therefore, the total cost \(C\) to rent a car is a function of the hours \(t\) the car is rented plus the flat fee.

    a.  Write the formula for the function that models this situation.

    b.  Find the total cost to rent a car for 2 days and 7 hours.

    c.  Determine how long the car was rented if the bill is $432.73.

    53) [T] A vehicle has a 20-gal tank and gets 15 mpg. The number of miles \(N\) that can be driven depends on the amount of gas \(x\) in the tank.

    a.  Write a formula that models this situation.

    b.  Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) 3/4 of a tank of gas.

    c.  Determine the domain and range of the function.

    d.  Determine how many times the driver had to stop for gas if she has driven a total of 578 mi.

    Answer:
    a. \(N(x)=15x\)
    b. i. \(N(20)=15(20)=300\); therefore, the vehicle can travel 300 mi on a full tank of gas.
    ii. \(N(15)=225\); therefore, the vehicle can travel 225 mi on 3/4 of a tank of gas.
    c. Domain: \(0≤x≤20\); range: \([0,300]\)
    d. The driver had to stop at least once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.

    54) [T] The volume \(V\) of a sphere depends on the length of its radius as \(V=(4/3)πr^3\). Because Earth is not a perfect sphere, we can use the mean radius when measuring from the center to its surface. The mean radius is the average distance from the physical center to the surface, based on a large number of samples. Find the volume of Earth with mean radius \(6.371×106\) m.

    55) [T] A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by \(r(t)=6−\dfrac{5}{t^2+1}\), where \(t\) is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture.

    a.  Express the area of the bacteria as a function of time.

    b.  Find the exact and approximate area of the bacterial culture in 3 hours.

    c.  Express the circumference of the bacteria as a function of time.

    d.  Find the exact and approximate circumference of the bacteria in 3 hours.

    Answer:
    a. \(A(t)=A(r(t))=π⋅(6−\frac{5}{t^2+1})^2\)
    b. Exact: \(\frac{121π}{4}\); approximately \(95\text{ cm}^2\)
    c. \(C(t)=C(r(t))=2π(6−\frac{5}{t^2+1})\)
    d. Exact: \(11π\); approximately \(35\) cm

    56) [T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function \(E(x)=0.79x\), where \(x\) is the number of U.S. dollars and \(E(x)\) is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is \(D(x)=1.245x\), where \(x\) is the number of Euros and \(D(x)\) is the equivalent number of U.S. dollars.

    a.  Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process?

    b.  Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she converted an extra $200 when she arrived in Paris.

    57) [T] The manager at a skateboard shop pays his workers a monthly salary \(S\) of $750 plus a commission of $8.50 for each skateboard they sell.

    a.  Write a function \(y=S(x)\) that models a worker’s monthly salary based on the number of skateboards \(x\) he or she sells.

    b.  Find the approximate monthly salary when a worker sells 25, 40, or 55 skateboards.

    c.  Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of $1400. (Hint: Find the intersection of the function and the line \(y=1400\).)

    An image of a graph. The y axis runs from 0 to 1800 and the x axis runs from 0 to 100. The graph is of the function “S(x) = 8.5x + 750”, which is a increasing straight line. The function has a y intercept at (0, 750) and the x intercept is not shown.

    Answer:
    a. \(S(x)=8.5x+750\) b. $962.50, $1090, $1217.50 c. 77 skateboards

    58) [T] Use a graphing calculator to graph the half-circle \(y=\sqrt{25−(x−4)^2}\). Then, use the INTERCEPT feature to find the value of both the \(x\)- and \(y\)-intercepts.

    An image of a graph. The y axis runs from -6 to 6 and the x axis runs from -1 to 10. The graph is of the function that is a semi-circle (the top half of a circle). The function has the begins at the point (-1, 0), runs through the point (0, 3), has maximum at the point (4, 5), and ends at the point (9, 0). None of these points are labeled, they are just for reference.

     

    Contributors

    Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


    1.1E: Exercises for Section 1.1 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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