For the pair of functions, (g⁰f)(x) is 4 x ² - 7 and (f ⁰g)(x) is 16 x ² + 8 x - 1
The given questions asks to find (g⁰f)(x) and (f ⁰g)(x), that is, function of g in terms of f(x) and function of f in terms of g(x).
To find : (g⁰f)(x)
Step 1 : Write g (x)
=> g (x) = 4 x + 1
Step 2 : Put f (x) in place of x in g (x)
=> g (f(x) = 4 ( x ² - 2 ) + 1
=> g (f(x) = 4 x ² - 8 + 1
=> g (f(x) = 4 x ² - 7
=> (g⁰f)(x) = 4 x ² - 7
Similarly,
To find : (f ⁰g)(x)
Step 1 : Write f (x)
=> f (x) = x ² - 2
Step 2 : Put g (x) in place of x in f (x)
=> f (g(x) = ( 4 x + 1 ) ² - 2
=> f (g(x) = 16 x ² + 1 + 8 x -2
=> f (g(x) = 16 x ² + 8 x - 1
=> (f ⁰g)(x) = 16 x ² + 8 x - 1
Therefore, we get for the pair of functions, (g⁰f)(x) as 4 x ² - 7 and (f ⁰g)(x) as 16 x ² + 8 x - 1
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