Example 1 :
If f(x) = 2x2 + 3 and g(x) = x + 2, find f o g.
Solution :
f o g = f[g(x)]
= f[x + 2]
= 2(x + 2)2 + 3
= 2(x2 + 4x + 4) + 3
= 2x2 + 8x + 8 + 3
= 2x2 + 8x + 11
Example 2 :
If f(x) = 5x and g(x) = x + 2, find f o g(x2).
Solution :
f o g(x2) = f[g(x2)]
= f[x2 + 2]
= 5(x2 + 2)
= 5x2 + 10
Example 3 :
If f(x) = 5x + 3 and g(x) = 7x - 2, find f o g(3).
Solution :
f o g(x) = f[g(x)]
= f[7x - 2]
= 5(7x - 2) + 3
= 35x - 10 + 3
f o g(x) = 35x - 7
f o g(3) = 35(3) - 7
= 105 - 7
= 98
Example 4 :
Using f(x) = 4x + 3 and g(x) = x - 2, find f[g(5)].
Solution :
f[g(x)] = f[x - 2]
= 4(x - 2) + 3
= 4x - 8 + 3
f[g(x)] = 4x - 5
f[g(5) = 4(5) - 5
= 20 - 5
= 15
Example 5 :
Using f(x) = 6x2 and g(x) = 14x + 4, find g o f.
Solution :
g o f = g[f(x)]
= g[6x2]
= 84x2 + 4
Example 6 :
Using f(x) = 5x + 4 and g(x) = x - 3, find g o f(6).
Solution :
g o f(x) = g[f(x)]
= g[5x + 4]
= (5x + 4) - 3
= 5x + 4 - 3
g o f(x) = 5x + 1
g o f(6) = 5(6) + 1
= 30 + 1
= 31
Example 7 :
If f(x) = x - 5 and g(x) = 2x + 3, verify f o g = g o f.
Solution :
f o g = [g(x)]
= f[2x + 3]
= (2x + 3) - 5
= 2x + 3 - 5
f o g = 2x - 2 ---->(1)
g o f = g[f(x)]
= g[x - 5]
= 2(x - 5) + 3
= 2x - 10 + 3
= 2x - 7 ---->(2)
From (1) and (2), we see that f o g ≠ g o f.
Example 8 :
Let f(x) = x + k and g(x) = 7x. If f o g(2) = 7, find k.
Solution :
f o g(2) = 7
f[g(2)] = 7
f[7(2)] = 7
f(14) = 7
14 + k = 7
k = -7
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