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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

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