Chapter 4.1 Practice

Questions

  1. Determine if the following functions are even, odd or neither.
    1. \(f(x)=x^2\cos(x)\)
    2. \(f(x)=x^3\cos(x)\)
    3. \(f(x)=x^4\sin(x)\)
    4. \(f(x)=x^7\sin(x)\)
    5. \(f(x)=x+\tan(x)\)
    6. \(f(x)=x^2+|x|\)
    7. \(f(x)=x|x|\)
    8. \(f(x)=x^3|x|+|x|\)
    9. \(f(x)=x^2|x|+|x|\)
  2. Identify a translation \(T\) so that for each function \(T(f)\) is either even or odd.
    1. \(f(x)=(x-5)^2+2\)
    2. \(f(x)=(x-3)|x-3|+1\)
    3. \(f(x)=x^3+2\)
    4. \(f(x)=\cos(x+3)+2\)
    5. \(f(x)=x\sin(x)-6\)
  3. For each function \(f\), determine a point \(p\) so that rotation around \(p\) by half a circle equals \(f\).
    1. \(f(x)=(x-3)|x-3|+1\)
    2. \(f(x)=x^3+2\)
    3. \(f(x)=\cos(x+3)+2\)
    4. \(f(x)=x\sin(x)-6\)
  4. For each function \(f\),determine a vertical line \(L\) so that reflection of \(f\) across \(L\) is equal to \(f.\)
    1. \(f(x)=(x-5)^2+2\)
    2. \(f(x)=\cos(x+3)+2\)
    3. \(f(x)=\cos(x)|x|-1\)
  5. Take \(f\) to be a function that is odd. Part of its graph is shown below. Sketch what \(f\) looks like for values of \(x\) that are negative.

  1. Take \(f\) to be a function that is even. Part of its graph is shown below. Sketch what \(f\) looks like for values of \(x\) that are negative.

Answers

    1. even
    2. odd
    3. odd
    4. even
    5. odd
    6. even
    7. odd
    8. neither
    9. even
    1. \(T=\langle -5,-2\rangle\) so that \(Tf\) is even
    2. \(T=\langle -3,-1\rangle\) so that \(Tf\) is odd
    3. \(T=\langle 0,-2\rangle\) so that \(Tf\) is odd
    4. \(T=\langle 3,-2\rangle\) so that \(Tf\) is even
    5. \(T=\langle 0,6\rangle\) so that \(Tf\) is odd
    1. \(p=(3,1)\)
    2. \(p=(0,2)\)
    3. \(p=(-3,2)\)
    4. \(p=(0,-6)\)
    1. \(x=5\)
    2. \(x=-3\)
    3. \(x=0\)

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