Given the functions

𝑓
(
𝑥
)
=
𝑥
2
+
4

𝑔
(
𝑥
)
=
𝑥
−
1
𝑥

Then
(
𝑓
∘
𝑔
)
(
𝑥
)
is equal to

Question

Given the functions

𝑓
(
𝑥
)
=
𝑥
2
+
4

𝑔
(
𝑥
)
=
𝑥
−
1
𝑥

Then 
(
𝑓
∘
𝑔
)
(
𝑥
)
 is equal to

BlackTom AI · Accepted Answer

The question asks us to find the composition (f ∘ g)(x) given f(x) = x^2 + 4 and g(x) = x − 1/x. To compute (f ∘ g)(x), we substitute g(x) into f: f(g(x)) = (g(x))^2 + 4.

First, evaluate g(x) explicitly: g(x) = x − 1/x.
Then square g(x): (g(x))^2 = (x − 1/x)^2 = x^2 − 2 + 1/x^2.
Add 4 to this result (from f): (f ∘ g)(x) = (x^2 − 2 + 1/x^2) + 4 = x^2 + 1/x^2 + 2.

Now, assess each answer option:

Option 1: 5x^2 − 2x + 1/x^2
- This expression would arise if there were a 5x^2 term and a −2x term, which does not occur from expanding (x − 1/x)^2 and adding 4. The cross term from squaring (x − 1/x) would be 0 for a −2x term, so this option misrepresents the interaction between x and 1/x. It also has x^2 and 1/x^2 with mismatched coefficients compared to the correct derivation.

Option 2: 5x^2 − 1/x^2
- Here, the constants and the 1/x^2 term are not accompanied by the +2 constant, and the 5x^2 coefficient is not supported by the expansion. The correct x^2 coefficient from the derivation is 1, not 5, and there is a +1/x^2 term with coefficient 1, not −1. This option fails to reflect the +2 constant and the proper combination of terms.

Option 3: x^2 + 2x + 1/x
- This option presents a linear term 2x and a first power 1/x, which do not appear in the expansion of (x − 1/x)^2 or in the final expression. There is no 2x or 1/x term produced by squaring x − 1/x and adding 4; the result consists of x^2, 1/x^2, and a constant 2.

Option 4: x^2 + 3x^2 + 4
- This option appears to combine x^2 terms into 4x^2 and then add a constant 4, but does not account for the 1/x^2 term at all and misrepresents the constant part. The correct expression includes 1/x^2 and has a constant 2, not 4.

Across all options, the common misconception would be misexpanding (x − 1/x)^2 or misadding the constant, leading to incorrect coefficients or missing terms. The precise expansion shows three components: x^2, 1/x^2, and a constant 2, with no linear term in x or in 1/x.

Putting it together, the computed (f ∘ g)(x) = x^2 + 1/x^2 + 2 does not match any of the provided answer choices. Therefore, none of the given options correctly represents (f ∘ g)(x) as derived above.

SH-GENMATH T-STEM11-O [SA-PT] Assignment 1 Function Definitions and Operations

Given the functions 𝑓 ( 𝑥 ) = 𝑥 2 + 4 𝑔 ( 𝑥 ) = 𝑥 − 1 𝑥 Then ( 𝑓 ∘ 𝑔 ) ( 𝑥 ) is equal to

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