Decomposition Review 1

\[ \definecolor{ucrblue}{rgb}{0.0627,0.3843,0.6039} \definecolor{ucrgold}{rgb}{0.9686,0.6941,0.2824} \definecolor{ucrred}{rgb}{0.8941,0,0.1686} \definecolor{ucrgreen}{rgb}{0.4706,0.7451,0.1255} \definecolor{ucraccent}{rgb}{1.0000,0.9569,0.8392} \DeclareMathOperator*{\LO}{O} \DeclareMathOperator*{\Lo}{o} \DeclareMathOperator*{\Recip}{Recip} \DeclareMathOperator*{\abs}{abs} \DeclareMathOperator{\pow}{pow} \]

Notes

You can reference these sections to review.

• Chapter 1.1: The Algebra of Sets
• Chapter 1.2: Intervals and Inequalities
• Chapter 1.3: Functions
• Chapter 1.4: Functions given by Simple Formulas
• Chapter 1.5: Manipulating Functions

Knowledge Checks

1. Take \(f(x)=x^3.\) Draw the function on the restriction \((-3,-1]\cup[1,\infty)\).
2. Take \(f\) and \(g\) to be functions with \(\mathcal{D}(f)=(-\infty,10]\), \(\mathcal{D}(g)=(-5,3]\cup (9,\infty)\), and the zero set of \(g\) is \(\{3,11\}\). Determine the domain of \(f+g\), \(fg\), and \(\frac{f}{g}\).
3. Determine how \(f-g\) is defined and the domain of \(f-g.\)
4. Take \(f\) and \(g\) to be given by \[f(x)=2x\quad\text{and}\quad g(x)=\sqrt{x-4}.\] Determine a formula for \(f\circ g\) and \(g\circ f\). Also determine the domain of both functions.
5. Take \[a(x)=x,\quad b(x)=x^2,\quad c(x)=x+3,\quad d(x)=5x,\quad\text{and}\quad e(x)=\sqrt{x}.\] Decompose \(f\) into sums, products, quotients and or composites of more elementary functions, where \[f(x)=x^2\sqrt{x+5x^2}+\frac{x+3}{x}.\]

• Knowledge Checks PDF
• Knowledge Checks Solutions

Remember, do the knowledge checks before checking the solutions.

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