SOLUTION: Algebra operations and composition of functions exam ... - Free Printable

Let’s solve each problem step by step. We’re doing function composition, which means plugging one function into another.

Remember:
- $ g(h(a)) $ means: take the output of $ h(a) $, and plug it into $ g $.
- $ (f \circ g)(x) $ is the same as $ f(g(x)) $ — again, plug $ g(x) $ into $ f $.

We’ll go one by one.

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1)
$ g(a) = 4a - 1 $
$ h(a) = a^2 + 5 $
Find $ g(h(a)) $

→ Plug $ h(a) $ into $ g $:
$ g(h(a)) = g(a^2 + 5) = 4(a^2 + 5) - 1 $
= $ 4a^2 + 20 - 1 $
= $ 4a^2 + 19 $

✔ Final Answer for #1: $ 4a^2 + 19 $

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2)
$ f(x) = 2x - 1 $
$ g(x) = -3x^2 - 2x $
Find $ (f \circ g)(x) = f(g(x)) $

→ Plug $ g(x) $ into $ f $:
$ f(-3x^2 - 2x) = 2(-3x^2 - 2x) - 1 $
= $ -6x^2 - 4x - 1 $

✔ Final Answer for #2: $ -6x^2 - 4x - 1 $

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3)
$ g(n) = -n - 3 $
$ h(n) = n^2 + 1 $
Find $ (g \circ h)(n) = g(h(n)) $

→ Plug $ h(n) $ into $ g $:
$ g(n^2 + 1) = -(n^2 + 1) - 3 $
= $ -n^2 - 1 - 3 $
= $ -n^2 - 4 $

✔ Final Answer for #3: $ -n^2 - 4 $

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4)
$ h(n) = -n - 1 $
$ g(n) = n^2 - n $
Find $ (h \circ g)(n) = h(g(n)) $

→ Plug $ g(n) $ into $ h $:
$ h(n^2 - n) = -(n^2 - n) - 1 $
= $ -n^2 + n - 1 $

✔ Final Answer for #4: $ -n^2 + n - 1 $

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5)
$ h(t) = -2t - 5 $
$ g(t) = -3t^2 - 2t $
Find $ h(g(t)) $

→ Plug $ g(t) $ into $ h $:
$ h(-3t^2 - 2t) = -2(-3t^2 - 2t) - 5 $
= $ 6t^2 + 4t - 5 $

✔ Final Answer for #5: $ 6t^2 + 4t - 5 $

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6)
$ f(a) = a - 2 $
$ g(a) = -3a^2 - 1 $
Find $ (f \circ g)(a) = f(g(a)) $

→ Plug $ g(a) $ into $ f $:
$ f(-3a^2 - 1) = (-3a^2 - 1) - 2 $
= $ -3a^2 - 3 $

✔ Final Answer for #6: $ -3a^2 - 3 $

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7)
$ g(x) = 2x - 5 $
$ f(x) = 4x + 2 $
Find $ g(f(x)) $

→ Plug $ f(x) $ into $ g $:
$ g(4x + 2) = 2(4x + 2) - 5 $
= $ 8x + 4 - 5 $
= $ 8x - 1 $

✔ Final Answer for #7: $ 8x - 1 $

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8)
$ h(x) = -3x - 3 $
$ g(x) = x^3 + 2x^2 $
Find $ (h \circ g)(x) = h(g(x)) $

→ Plug $ g(x) $ into $ h $:
$ h(x^3 + 2x^2) = -3(x^3 + 2x^2) - 3 $
= $ -3x^3 - 6x^2 - 3 $

✔ Final Answer for #8: $ -3x^3 - 6x^2 - 3 $

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9)
$ g(a) = a^2 + 3 $
$ h(a) = 4a + 4 $
Find $ g(h(a)) $

→ Plug $ h(a) $ into $ g $:
$ g(4a + 4) = (4a + 4)^2 + 3 $
First expand: $ (4a + 4)^2 = 16a^2 + 32a + 16 $
Then add 3: $ 16a^2 + 32a + 16 + 3 = 16a^2 + 32a + 19 $

✔ Final Answer for #9: $ 16a^2 + 32a + 19 $

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10)
$ f(x) = x^2 + 5 $
$ g(x) = 3x + 3 $
Find $ f(g(x)) $

→ Plug $ g(x) $ into $ f $:
$ f(3x + 3) = (3x + 3)^2 + 5 $
Expand: $ (3x + 3)^2 = 9x^2 + 18x + 9 $
Add 5: $ 9x^2 + 18x + 9 + 5 = 9x^2 + 18x + 14 $

✔ Final Answer for #10: $ 9x^2 + 18x + 14 $

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Final Answer:
1) $ 4a^2 + 19 $
2) $ -6x^2 - 4x - 1 $
3) $ -n^2 - 4 $
4) $ -n^2 + n - 1 $
5) $ 6t^2 + 4t - 5 $
6) $ -3a^2 - 3 $
7) $ 8x - 1 $
8) $ -3x^3 - 6x^2 - 3 $
9) $ 16a^2 + 32a + 19 $
10) $ 9x^2 + 18x + 14 $