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3.2.1: Exercises 3.2

  • Page ID
    63464
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    Terms and Concepts

    Exercise \(\PageIndex{1}\)

    In which situations is substitution a more appropriate solution method than equating the functions?

    Answer

    When one or both functions are defined implicitly.

    Exercise \(\PageIndex{2}\)

    Is \(y=x^3+5x-7\) an implicitly or explicitly defined function? Explain.

    Answer

    Explicitly; \(y\) is isolated.

    Exercise \(\PageIndex{3}\)

    Is \(xy+y^2 -y = 2x+6\) an implicitly or explicitly defined function? Explain.

    Answer

    Implicitly; \(y\) is not isolated.

    Exercise \(\PageIndex{4}\)

    Describe the pros and cons of using graphing to find the point(s) of intersection.

    Answer

    Answers will vary, but graphing helps you determine how many intersections points exist, but does not always clearly show the exact values.

    Problems

    In exercises \(\PageIndex{5}\) - \(\PageIndex{8}\), determine the maximum possible number of intersections for the described functions.

    Exercise \(\PageIndex{5}\)

    Two linear functions with different slopes

    Answer

    1

    Exercise \(\PageIndex{6}\)

    A linear function and a quadratic function

    Answer

    2

    Exercise \(\PageIndex{7}\)

    Two explicitly defined quadratic functions

    Answer

    2

    Exercise \(\PageIndex{8}\)

    A cubic function and a constant function

    Answer

    3

    In exercises \(\PageIndex{9}\) - \(\PageIndex{12}\), determine the minimum possible number of intersections for the described functions.

    Exercise \(\PageIndex{9}\)

    Two linear functions with different slopes

    Answer

    1

    Exercise \(\PageIndex{10}\)

    A linear function and a quadratic function

    Answer

    0

    Exercise \(\PageIndex{11}\)

    Two explicitly defined quadratic functions

    Answer

    0

    Exercise \(\PageIndex{12}\)

    A cubic function and a constant function

    Answer

    1

    In exercises \(\PageIndex{13}\) - \(\PageIndex{18}\), find all points of intersection between the given functions.

    Exercise \(\PageIndex{13}\)

    \(y=x^2-1\) and \(y=x-1\)

    Answer

    \((0,-1)\) and \((1,0)\)

    Exercise \(\PageIndex{14}\)

    \(x^2+y^2=1\) and \(4y=3x\)

    Answer

    \((\frac{4}{5},\frac{3}{5})\) and \((-\frac{4}{5}, -\frac{3}{5})\)

    Exercise \(\PageIndex{15}\)

    \(y-1 = \sqrt{3x}\) and \(y=x+1\)

    Answer

    \((0,1)\) and \((3,4)\)

    Exercise \(\PageIndex{16}\)

    \(y=x^2-3x+2\) and the x-axis

    Answer

    \((1,0)\) and \((2,0)\)

    Exercise \(\PageIndex{17}\)

    \(y=x^2-3x+2\) and \(y=5\)

    Answer

    \((\frac{3+\sqrt{21}}{2},5)\) and \((\frac{3-\sqrt{21}}{2},5)\)

    Exercise \(\PageIndex{18}\)

    \(y+2x=5\) and \(y+3=x^3-7x^2+12x\)

    Answer

    \((1,3)\), \((2,1)\), and \((4,-3)\)

    In exercises \(\PageIndex{19}\) - \(\PageIndex{22}\), sketch the region bounded by the given functions and determine all intersection points.

    Exercise \(\PageIndex{19}\)

    \(y=x^2\) and \(y=x\)

    Answer

    Points of intersection are \((0,0)\) and \((1,1)\)

    Exercise \(\PageIndex{20}\)

    \(y=x^2\) and \(y=x+2\)

    Answer

    Points of intersection are \((-1,1)\) and \((2,4)\)

    Exercise \(\PageIndex{21}\)

    \(y=x^2\) and \(y=\sqrt{x}\)

    Answer

    Points of intersection are \((0,0)\) and \((1,1)\)

    Exercise \(\PageIndex{22}\)

    \(3y+2x=6\), the x-axis, and the y-axis (hint: sketch before looking for the intersection points)

    Answer

    Points of intersection are \((0,0)\), \((0,2)\), and \((3,0)\)


    3.2.1: Exercises 3.2 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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