1.2.1: Exercises 1.2
- Page ID
- 62179
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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 \(\PageIndex{1}\)
What does “the function \(f(t)\)” tell you that “the function \(f\)” does not?
- Answer
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It tells us that \(t\) is the input for the function \(f\).
Exercise \(\PageIndex{2}\)
T/F: If \(g(x)=x^2\), then \(g(2)=g(-2)\).
- Answer
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True
Exercise \(\PageIndex{3}\)
T/F: You can’t combine functions using both composition and quotients in the same function.
- Answer
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False
Exercise \(\PageIndex{4}\)
T/F: In the combination \(g(f(x))\), \(f(x)\) is the input for \(g(x)\).
- Answer
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True
Problems
Let \(f(x) = x^{3}\), \(g(x) = x + 4\), and \(h(x) = \sin (x)\). Each of exercise \(\PageIndex{5}\) – \(\PageIndex{8}\) is some combination of \(f(x)\), \(g(x)\), and \(h(x)\). Determine the type of combination and write it using function notation. For example, \(x^{3} + x + 4\) is the addition of \(f(x)\) and \(g(x)\) and can be written as \(f(x) + g(x)\).
Exercise \(\PageIndex{5}\)
\(\frac{x^3}{\sin{(x)}}\)
- Answer
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Quotient of \(f(x)\) and \(h(x)\); \(\frac{f(x)}{h(x)}\)
Exercise \(\PageIndex{6}\)
\(\sin{(x+4)}\)
- Answer
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Composition of \(h(x)\) with \(g(x)\); \(h(g(x))\)
Exercise \(\PageIndex{7}\)
\(\sin{(x)}+4\)
- Answer
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Composition of \(g(x)\) with \(h(x)\); \(g(h(x))\)
Exercise \(\PageIndex{8}\)
\((2x^3)(x+4)\)
- Answer
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Product of a scalar multiple of \(f(x)\) with \(g(x)\); \((2f(x))(g(x))\)
In exercises \(\PageIndex{9}\) – \(\PageIndex{11}\), determine the input variable of each function, any parameters of the function, and the type of function.
Exercise \(\PageIndex{9}\)
\(C(A) = \frac{k \epsilon_0 A}{d}\)
- Answer
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The input variable is \(A\). The parameters are \(k\), \(\epsilon_0\), and \(d\). This is a monomial of degree 1.
Exercise \(\PageIndex{10}\)
\(v(t) = -9.8t + v_0\)
- Answer
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The input variable is \(t\). The only parameter is \(v_0\). This is a polynomial of degree 1.
Exercise \(\PageIndex{11}\)
\(A(t)=P(1+\frac{r}{n})^{nt}\)
- Answer
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The input variable is \(t\). The parameters are \(P\), \(r\), and \(n\). This is an exponential function.
In exercises \(\PageIndex{12}\) – \(\PageIndex{17}\), evaluate the given expression.
Exercise \(\PageIndex{12}\)
Given \(f(x)=2x^2\) and \(g(x)=x-b\), find \(5f(3a)-g(4)\)
- Answer
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\(90a^2-4+b\)
Exercise \(\PageIndex{13}\)
Given \(f(x)=x^2-3\) and \(g(x)=x-b\), find \(f(y+h)-3g(5)\)
- Answer
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\(y^2+2yh+h^2-18+3b\)
Exercise \(\PageIndex{14}\)
Given \(f(x)=5-x\) and \(g(x)=-x^4+p\), find \(f(y+h)-3g(y)\)
- Answer
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\(5-y-h+3y^4-3p\)
Exercise \(\PageIndex{15}\)
Given \(f(\theta)=\frac{\theta+3}{\theta-2}\) and \(g(\theta)=\theta^2+4\), find \(g(f(3))\)
- Answer
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\(40\)
Exercise \(\PageIndex{16}\)
Given \(g(x)=x^2-4\) and \(f(x)=\sqrt{x+8}\), find \(g(x+h)-2f(8)\)
- Answer
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\(x^2+2xh+h^2-12\)
Exercise \(\PageIndex{17}\)
Given \(f(y)=y-5\) and \(g(y)=h-y^2\), find \(g(f(y))-f(g(y))\)
- Answer
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\(10y-20\)
In exercises \(\PageIndex{18}\) – \(\PageIndex{21}\), determine the difference quotient of each of the following functions.
Exercise \(\PageIndex{18}\)
\(h(r)=2r+4\)
- Answer
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\(2\)
Exercise \(\PageIndex{19}\)
\(g(y)=4y-7\)
- Answer
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\(4\)
Exercise \(\PageIndex{20}\)
\(y(x)=x^2+6\)
- Answer
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\(2x+h\)
Exercise \(\PageIndex{21}\)
\(f(t)=4t^2+x\)
- Answer
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\(8t+4h\)