If f : R → R is given by f(x) = (5 + x4)1/4, then fοf(x) is _______a)xb)10 + x4c)5 + x4d)(10 + x4)1/4Correct answer is option 'D'. Can you explain this answer?

Question

If f : R &rarr; R is given by f(x) = (5 + x4)1/4, then f&omicron;f(x) is _______a)xb)10 + x4c)5 + x4d)(10 + x4)1/4Correct answer is option 'D'. Can you explain this answer?

Answer

Explanation:

Composition of Functions:
- When we compose two functions f and g, denoted as f∘g(x), it means we first apply g to x and then apply f to the result.
- Mathematically, f∘g(x) = f(g(x))

Given Function:
- The given function is f(x) = (5 + x^4)^(1/4)

Find f∘f(x):
- Substituting f(x) into f∘f(x), we get f∘f(x) = f(f(x))
- f(f(x)) = f((5 + x^4)^(1/4))
- Since f(x) = (5 + x^4)^(1/4), we substitute this into the expression above:
f((5 + x^4)^(1/4)) = ((5 + (5 + x^4)^(1/4)^4)^(1/4))
= ((5 + (5 + x^4))^(1/4))
= ((10 + x^4)^(1/4))

Therefore, the correct answer is option 'D': (10 + x^4)^(1/4)

Answer

Given that f(x) = (5 + x⁴)^1/4
∴ fοf(x) = f(f(x)) = (5 + {(5 + x⁴)^1/4}⁴)^1/4
= (5 + (5 + x⁴))^1/4= (10+x⁴)^1/4

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