3.2.1: Exercises 3.2
- Page ID
- 63464
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Terms and Concepts
Exercise \(\PageIndex{1}\)
In which situations is substitution a more appropriate solution method than equating the functions?
- Answer
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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
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Explicitly; \(y\) is isolated.
Exercise \(\PageIndex{3}\)
Is \(xy+y^2 -y = 2x+6\) an implicitly or explicitly defined function? Explain.
- Answer
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Implicitly; \(y\) is not isolated.
Exercise \(\PageIndex{4}\)
Describe the pros and cons of using graphing to find the point(s) of intersection.
- Answer
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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
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1
Exercise \(\PageIndex{6}\)
A linear function and a quadratic function
- Answer
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2
Exercise \(\PageIndex{7}\)
Two explicitly defined quadratic functions
- Answer
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2
Exercise \(\PageIndex{8}\)
A cubic function and a constant function
- Answer
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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
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1
Exercise \(\PageIndex{10}\)
A linear function and a quadratic function
- Answer
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0
Exercise \(\PageIndex{11}\)
Two explicitly defined quadratic functions
- Answer
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0
Exercise \(\PageIndex{12}\)
A cubic function and a constant function
- Answer
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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
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\((0,-1)\) and \((1,0)\)
Exercise \(\PageIndex{14}\)
\(x^2+y^2=1\) and \(4y=3x\)
- Answer
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\((\frac{4}{5},\frac{3}{5})\) and \((-\frac{4}{5}, -\frac{3}{5})\)
Exercise \(\PageIndex{15}\)
\(y-1 = \sqrt{3x}\) and \(y=x+1\)
- Answer
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\((0,1)\) and \((3,4)\)
Exercise \(\PageIndex{16}\)
\(y=x^2-3x+2\) and the x-axis
- Answer
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\((1,0)\) and \((2,0)\)
Exercise \(\PageIndex{17}\)
\(y=x^2-3x+2\) and \(y=5\)
- Answer
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\((\frac{3+\sqrt{21}}{2},5)\) and \((\frac{3-\sqrt{21}}{2},5)\)
Exercise \(\PageIndex{18}\)
\(y+2x=5\) and \(y+3=x^3-7x^2+12x\)
- Answer
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\((1,3)\), \((2,1)\), 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
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Points of intersection are \((0,0)\) and \((1,1)\)
Exercise \(\PageIndex{20}\)
\(y=x^2\) and \(y=x+2\)
- Answer
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Points of intersection are \((-1,1)\) and \((2,4)\)
Exercise \(\PageIndex{21}\)
\(y=x^2\) and \(y=\sqrt{x}\)
- Answer
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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
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Points of intersection are \((0,0)\), \((0,2)\), and \((3,0)\)