Answers
The composite functions are listed below:
g ° f(3) = 29 g ° f(x + 1) = 2 · x² - 1 h ° g (- 3 · x) = - 36 · x - 36 f ° g (- 2 - t) = 73 + 72 · t + 16 · t² f ° g (x²) = 3 · x² + 17How to evaluate composite functions
In this problem we find five cases of composite functions, each of which has to be evaluated. A composition between the two functions f and g is defined below:
f ° g(x) = f ( g(x) )
Now we proceed to evaluate each of the five cases by using algebra properties and the definition of the composition of functions:
Case 1
g ° f(3) = g( f(3) )
g ° f(3) = 4 · (2 · 3 + 2) - 3
g ° f(3) = 4 · 8 - 3
g ° f(3) = 29
Case 2
g ° f(x + 1) = g( f(x + 1) )
g ° f(x + 1) = 2 · [(x + 1)² - 2 · (x + 1)] + 1
g ° f(x + 1) = 2 · (x² + 2 · x + 1 - 2 · x - 2) + 1
g ° f(x + 1) = 2 · (x² - 1) + 1
g ° f(x + 1) = 2 · x² - 1
Case 3
h ° g (- 3 · x) = h( g (- 3 · x) )
h ° g (- 3 · x) = 3 · [4 · (- 3 · x) - 3]
h ° g (- 3 · x) = 3 · (- 12 · x - 12)
h ° g (- 3 · x) = - 36 · x - 36
Case 4
f ° g (- 2 - t) = f( g (- 2 - t))
f ° g (- 2 - t) = [4 · (- 2 - t) - 2]² + 1 + 2 · [4 · (- 2 - t) - 2]
f ° g (- 2 - t) = (- 8 - 4 · t - 2)² + 1 + 2 · (- 8 - 4 · t - 2)
f ° g (- 2 - t) = (- 10 - 4 · t)² + 1 + 2 · (- 10 - 4 · t)
f ° g (- 2 - t) = 100 + 80 · t + 16 · t² + 1 - 20 - 8 · t
f ° g (- 2 - t) = 73 + 72 · t + 16 · t²
Case 5
f ° g (x²) = f( g(x) )
f ° g (x²) = 3 · (x² + 4) + 5
f ° g (x²) = 3 · x² + 12 + 5
f ° g (x²) = 3 · x² + 17
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--------------------
1. Kris
2. Katie
---------------------
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