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Question
Answer the following:
Find f(x) if g(x) = x2 + x – 2 and (g ° f) (x) = 4x2 – 10x + 4
Solution
g(x) = x2 + x – 2 and (g ° f) (x) = 4x2 – 10x + 4
4x2 – 10x + 4 = g[f(x)] = [f(x)]2 + [f(x)] – 2
∵ deg(4x2 – 10x + 4) = 2
∴ deg [f(x)] = 1
Let f(x) = ax + b, a, b ∈ R
∴ 4x2 – 10x + 4 = (ax + b)2 + (ax + b) – 2
= a2x2 + 2abx + b2 + ax + b – 2
= a2x2 + (2ab + a)x + b2 + b – 2
Comparing the corresponding coefficients, we get
a2 = 4 ...(1)
2ab + a = – 10 ...(2)
b2 + b – 2 = 4 ...(3)
From (1), a = ± 2
If a = 2, from (2), 4b + 2 = – 10
∴ b = – 3
b = – 3 satisfies b2 + b – 2 = 4
∴ f(x) = 2x – 3 is one possibility
If a = – 2, from (2), – 4b – 2 = – 10
∴ b = 2
b = 2 satisfies (3)
∴ f(x) = – 2x + 2 is other possibility
∴ f(x) = 2x – 3 or f(x) = – 2x + 2
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