Decomposition Review 2

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

Notes

You can reference these sections to review.

• Chapter 1.6: Manipulating Functions

Knowledge Checks

1. Take \(f\) to be the function given by \[ f(x)= \begin{cases} 3&\text{if } -3\leq x<-1\\ \left(\frac{1}{2}\right)^{x+1}+1&\text{if } -1\leq x<1\\ -2x+3&\text{if }1\leq x<3. \end{cases} \] Identify its domain.
2. Take \(f\) to be the function given by \[ f(x)= \begin{cases} 3&\text{if } -3\leq x<-1\\ \left(\frac{1}{2}\right)^{x+1}+1&\text{if } -1\leq x<1\\ -2x+3&\text{if }1\leq x<3. \end{cases} \] Sketch \(f\).
3. Take \(f\) to be the piecewise function whose graph is given below.

It is a piecewise function made up of linear and quadratic functions. Determine a formula for \(f\).

4. Take \(f\) and \(g\) to be the functions given by \[f(x)=\begin{cases}x-3 &\text{ if } x\leq 4\\-x+9 &\text{ if } x> 4\end{cases}\quad\text{and}\quad g(x)=\begin{cases}3x-2&\text{ if } x<6\\ 3x-16 &\text{ if } x\geq 6.\end{cases}\] Solve the inequality \(f(x)>g(x)\) without using graphing software. Write your answer as a union of intervals.

5. Take \(f\) to be the function given by \[f(x)=\begin{cases}3x&\text{if }x\leq 2\\ -x^2&\text{if }4<x\leq 5\end{cases}\] and \(g\) to be the piecewise linear function whose graph is given below.

Write \(f\circ g\) as a piecewise function and state its domain.

6. Take \(f\) and \(g\) to be functions whose graphs are given below. Solve the inequality \(f(x)g(x)\geq0\).

7. Take \(f\) and \(g\) to be functions whose graphs are given below. Solve the inequality \(f(x)g(x)<0\).

• Knowledge Checks PDF
• Knowledge Checks Solutions

Remember, do the knowledge checks before checking the solutions.

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