FunctionMedium
Question
If g {f(x)} =| sin x | and f{g(x)} = (sin √x)2, then
Options
A.f(x) = sin2 x, g(x) = √x
B.f(x) = sin x, g(x) = | x |
C.f(x) = x2 , g(x) = √x
D.f and g cannot be determined
Solution
Let f (x) = sin2 x and g (x) = √x
Now, fog(x) = f[g(x)] = f(√x) = sin2 √x
and gof(x) = g[f(x)] = g(sin2x)
= |sin x|
Again let f(x) = sin x, g(x) = |x|
fog (x) = f[g(x)] = f(|x|) = sin |x| ≠ (sin √x)2
When f(x) = xx, g(x) = sin √x
fog(x) = f[g(x)] = f(sin √x) = (sin √x)2
and (gof)(x) = g[f(x)] = g(x2) = sin √x2
= sin |x| ≠ |sin x|
Therefore, (a) is the answer.
Now, fog(x) = f[g(x)] = f(√x) = sin2 √x
and gof(x) = g[f(x)] = g(sin2x)
= |sin x|Again let f(x) = sin x, g(x) = |x|
fog (x) = f[g(x)] = f(|x|) = sin |x| ≠ (sin √x)2
When f(x) = xx, g(x) = sin √x
fog(x) = f[g(x)] = f(sin √x) = (sin √x)2
and (gof)(x) = g[f(x)] = g(x2) = sin √x2
= sin |x| ≠ |sin x|
Therefore, (a) is the answer.
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