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

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# Composite Functions: Domain (Grades 11-12)

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## Composite Functions: Domain

1.
Find $(f@g)(x)$ and state its domain if $f(x) = 3x^2 + 5x - 3$ and $g(x) = 4x-1$.
1. $48x^2 - 4x - 5, \ \ RR$
2. $48x^2 - 19x, \ \ RR$
3. $3x^2 + 20x-8, \ \ x!=1/4$
4. $48x^2 - 4x - 5, \ \ x!= 1/4$
2.
Find $(g@f)(x)$ and state its domain if $f(x) = (3x-5)/(9x-1)$ and $g(x) = 3x + 4$.
1. $(9x+7) / (27x + 35), \ \ x!= -27/35$
2. $(9x+7) / (9x - 1), \ \ x!= 1/9$
3. $(9x+7) / (27x+35), \ \ x!= -35/27$
4. $(45x-19) / (9x-1), \ \ x!=1/9$
3.
If $f(x) = x^2 + 3$ and $g(x) = sqrt(x-4)$ find $(f@g)(x)$ and state its domain.
1. $x-1, \ \ x >= 4$
2. $sqrt(x^2 - 1), \ \ x < -1 or x > 1$
3. $x+1, \ \ x >= 4$
4. $x+1, \ \RR$
4.
For $f(x) = sqrt(x-1)$ and $g(x) = x^2 + 5$, find $(f@g)(x)$ and state its domain.
1. $|x -1| + 4, \ \ RR$
2. $sqrt(x^2 + 4), \ \ RR$
3. $sqrt(x^2 + 4), \ \ -2 <= x <= 2$
4. $sqrt(x^2 + 4), \ \ x >= 1$
5.
Find $(f@g)(x)$ and state its domain if $f(x) = sqrt(x + 8)$ and $g(x) = 2x^2 - 12$.
1. $sqrt(2x^2 - 4), \ \ x>=sqrt(2)$
2. $sqrt(2x^2 - 4), \ \ x <= -sqrt(2) or x >= sqrt(2)$
3. $sqrt(2x^2 - 4), \ \ x>=2$
4. $sqrt(x-4), \ \ x>=4$
6.
For $f(x) = e^(x + 3)$ and $g(x) = 1/(x-2)$, find $(f@g)(x)$ and state its domain.
1. $e^(x+1), \ \ RR$
2. $e^(1 \ // \ (x+1)), \ \ x != -1$
3. $e^((3x-5) \ // \ (x-2)), \ \ x != 2$
4. $1/(e^(x+3) - 2), \ \ x != ln(2) -3$
7.
If $f(x) = 4x^2 - 3x - 7$ and $g(x) = ln(x + 6)$, find $(g@f)(x)$ and its domain.
1. $ln(4x^2 - 3x - 1), \ \ x > -6$
2. $ln(4x^2 - 3x - 1), \ \ x > 1$
3. $ln(4x^2 - 3x - 1), \ \ x < -1/4 or x > 1$
4. $ln(4x^2 - 3x + 1), \ \ RR$
8.
Find $(f@g)(x)$ and its domain if $f(x) = x^3 + 2x^2 - 8x + 1$ and $g(x) = root[3](x)$.
1. $x + 2root[3](x^2) - 8root[3](x) + 1, \ \ RR$
2. $x + 2root[2](x^3) - 8root[3](x) + 1, \ \ x >= 0$
3. $x + 2x^2 - 8x + 1, \ \ RR$
4. $x + 2root[3](x^2) - 8root[3](x) + 1, \ \ x >= 0$
9.
If $f(x) = (8x+1) / (10x-5)$ and $g(x) = sqrt(x-2)$, find $(f@g)(x)$, simplified, and state its domain.
1. $(8sqrt(x-2)+1) / (5(2x-1)), \ \ [2,oo)$
2. $(8sqrt(x-2)+1) / (5(2x-1)), \ \ (-oo,1/2)uu(1/2,oo)$
3. $(16x + 10 sqrt(x-2) - 31) / (5(4x-9)), \ \ [2,9/4) uu (9/4,oo)$
4. $(16x + 10 sqrt(x-2) - 31) / (5(4x-9)), \ \ (-oo,9/4) uu (9/4,oo)$
10.
Find $(f@g)(x)$ and state its domain, if $f(x) = log(-x-5)$ and $g(x) = sqrt(x)$.
1. $log(-sqrt(x)-5), \ \ x>0$
2. $log(-sqrt(x)-5), \ \ x<0$
3. $log(-sqrt(x)-5), \ \ x>25$
4. This composite function is not possible.
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