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This printable supports Common Core Mathematics Standard HSF-BF.A.1, HSF-BF.A.1c

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Decomposing Functions (Grades 11-12)

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Decomposing Functions

1. 
If [math](f@g)(x) = 10x - 2[/math], which of the following are possible function rules for [math]f(x)[/math] and [math]g(x) ?[/math] Choose all correct answers.
  1. [math]f(x) = 10x + 10, \ \ g(x) = x - 12[/math]
  2. [math]f(x) = 2x-4, \ \ g(x) = 5x+1[/math]
  3. [math]f(x) = 8x-9, \ \ g(x) = 5/4 x - 11/4[/math]
  4. [math]f(x) = 1/2x - 3 \ \ g(x) = 20x + 2[/math]
2. 
If [math](f@g)(x) = x^2 + 2x + 4[/math], which of the following could be the functions [math]f(x)[/math] and [math]g(x) ?[/math] Choose all correct answers.
  1. [math]f(x) = x^2 + 3, \ \ g(x) = x+1[/math]
  2. [math]f(x) = x + 2, \ \ g(x) = x^2 + 2x + 2[/math]
  3. [math]f(x) = x - 5, \ \ g(x) = x^2 + 2x + 9[/math]
  4. [math]f(x) = x^2 + 2, \ \ g(x) = 2x[/math]
3. 
If [math](f@g)(x) = 1/(x^2 + 2)[/math], what could be the possible function rules for [math]f(x)[/math] and [math]g(x) ?[/math] There may be more than one correct answer.
  1. [math]f(x) = 1/(x^2 + 1), \ \ g(x) = x+1[/math]
  2. [math]f(x) = 1/(sqrt(x)), \ \ g(x) =(x^2+2)^2[/math]
  3. [math]f(x) = 1/(x^2 + 2), \ \ g(x) = x[/math]
  4. [math]f(x) = 1/(x-1), \ \ g(x) = x^2 + 3[/math]
4. 
What are possible function rules for [math]f(x)[/math] and [math]g(x)[/math], if [math](g@f)(x) = 9x^2 - 15x -13 ?[/math] There may be more than one correct answer.
  1. [math]f(x) = x-2, \ \ g(x) = 9x^2+21x-7[/math]
  2. [math]f(x) = 3x+2, \ \ g(x) = x^2 - 9x + 1[/math]
  3. [math]f(x) = 3x^2 -5x - 5, \ \ g(x) = 3x-2[/math]
  4. [math]f(x) = 3x^2 - 5x - 13, \ \ g(x) = 3x[/math]
5. 
For [math](f@g)(x) = e^(-x^2 + 6)[/math], which of the following are possible function rules of [math]f(x)[/math] and [math]g(x) ?[/math] There may be more than one correct answer.
  1. [math]f(x) = e^x, \ \ g(x) = -x^2 + 6[/math]
  2. [math]f(x) = e^(-x+6), \ \ g(x) = x^2[/math]
  3. [math]f(x) = e^(-x^2 - 4x + 2), \ \ g(x) = x-2[/math]
  4. [math]f(x) = x^2, \ \ g(x) = e^(-1/2x^2 + 3)[/math]
6. 
Which of the following could be the function rules for [math]f(x)[/math] and [math]g(x)[/math] if [math](f@g)(x) = sqrt(x^3-8) ?[/math] There may be more than one correct answer.
  1. [math]f(x) = sqrt(x), \ \ g(x) = x^2 - 8[/math]
  2. [math]f(x) = sqrt(x^2-8), \ \ g(x) = x[/math]
  3. [math]f(x) = sqrt(x-8), \ \ g(x) = x^3[/math]
  4. [math]f(x) = sqrt(x^3), \ \ g(x) = x-2[/math]
7. 
If [math](f@g)(x) = x^4 - 6x^2 + 7x + 9[/math], which of the following are possible functions rules for [math]f(x)[/math] and [math]g(x) ?[/math] There may be more than one correct answer.
  1. [math]f(x) = x^2 + 7x, \ \ g(x) = x^2 - 3[/math]
  2. [math]f(x) = x^2, \ \ g(x) = x^2 +7/3 x + 3[/math]
  3. [math]f(x) = x+3, \ \ g(x) = x^4 - 6x^2 + 7x + 6[/math]
  4. [math]f(x) = x^4 + x^3 - 17/4, \ \ g(x) = x - 1/4[/math]
8. 
Select the possible function rules of [math]f(x)[/math] and [math]g(x)[/math] if [math](f@g)(x) = x^2 - 4[/math]. Choose all correct answers.
  1. [math]f(x) = ln(x), \ \ g(x) = e^(x^2-4)[/math]
  2. [math]f(x) = ln(x^2), \ \ g(x) = e^(1/2 x^2 -2)[/math]
  3. [math]f(x) = ln(x^2 - 8), \ \ g(x) = e^(1/2 x^2 + 2)[/math]
  4. [math]f(x) = e^x - 8, \ \ g(x) = ln(x^2+4)[/math]
9. 
If [math](f@g)(x) = x[/math], which of the following could be possible function rules for [math]f(x)[/math] and [math]g(x) ?[/math] Choose all correct answers.
  1. [math]f(x) = 1/2sqrt(x), \ \ g(x) = 2x^2[/math]
  2. [math]f(x) = 1/4x - 1, \ \ g(x) = 4x+4[/math]
  3. [math]f(x) = ln(sqrt(x)) -2, \ \ g(x) = e^(2x+4)[/math]
  4. [math]f(x) = root[3](x-2), \ \ g(x) = (x+2)^3[/math]
10. 
If [math](f@g)(x) = 2|x+6|[/math], what are possible function rules for [math]f(x)[/math] and [math]g(x) ?[/math] There may be more than one correct answer.
  1. [math]f(x) = sqrt(x), \ \ g(x) = 4x^2 + 48x + 144[/math]
  2. [math]f(x) = sqrt(x^2 + 6), \ \ g(x) = 2x[/math]
  3. [math]f(x) = 4x^2 + 6, \ \ g(x) = 1/2sqrt(x)[/math]
  4. [math]f(x) = x^2 + 6, \ \ g(x) = 2sqrt(x)[/math]

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