If we define the functions and that map into by
Duration: 3 min
2014
If we define the functions f, g and h that map R into R by :
\(f(x) = x^4, g(x) = \sqrt {x^2 + 1}, h(x) = x^2 + 72\) , then the value of the composite functions ho(gof) and (hog)of are given as
- A.
\(x^8 - 71 \ \ and \ \ x^8 -71\) - B.
\(x^8 - 73 \ \ and \ \ x^8 -73\) - C.
\(x^8 + 71 \ \ and \ \ x^8 +71\) - D.
\(x^8 + 73 \ \ and \ \ x^8 +73\)
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Correct answer: D
Final result: both composite functions equal x^8 + 73.
Reason:
Start with f(x) = x^4.
Compute g(f(x)): g(f(x)) = sqrt((f(x))^2 + 1) = sqrt((x^4)^2 + 1) = sqrt(x^8 + 1).
Apply h to that result: h(g(f(x))) = (sqrt(x^8 + 1))^2 + 72 = x^8 + 1 + 72 = x^8 + 73.
Note:
Because function composition is associative, h∘(g∘f) = (h∘g)∘f, so both given composite expressions produce the same final formula.
The constant 73 comes from 1 (inside g) plus 72 (added by h).
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