If f(x) = 2x + 1 and g(x) = x² - 3, which of the following represents f(g(x))?

Rationale

f(g(x)) is represented by 2x² - 2.

To find f(g(x)), we substitute g(x) into f(x). Given f(x) = 2x + 1 and g(x) = x² - 3, we replace x in f with g(x) to calculate f(g(x)) = 2(g(x)) + 1, resulting in 2(x² - 3) + 1 = 2x² - 6 + 1 = 2x² - 5.

A (2x² - 2) CORRECT

This choice incorrectly simplifies the expression from f(g(x)). The correct substitution leads to f(g(x)) = 2(x² - 3) + 1, which simplifies to 2x² - 5, not 2x² - 2.

B (2x² + 1) YOUR ANSWER

This option fails to account for the substitution of g(x) into f(x). It suggests that f(g(x)) results in a linear transformation instead of correctly applying the input from g(x). The true computation shows that the output is significantly impacted by the -3 from g(x).

C (2x² - 5) YOUR ANSWER

This is actually the correct representation of f(g(x)). Upon substituting g(x) into f(x) and simplifying, we indeed find f(g(x)) = 2(x² - 3) + 1 = 2x² - 6 + 1 = 2x² - 5.

D (2x² - 3) YOUR ANSWER

This choice ignores the constant term introduced by the substitution. The transformation of g(x) does not directly lead to -3 as a result, as the full expression of f(g(x)) includes the adjustment from the -3 in g(x) plus the constant from f(x).

Conclusion

The correct evaluation of f(g(x)) leads to the expression 2x² - 5, as derived from substituting g(x) into f(x). The other options miscalculate or misinterpret the function transformations. Understanding how to properly substitute and simplify composite functions is crucial in achieving the correct result.

If f(x) = 2x + 1 and g(x) = x² - 3, which of the following represents f(g(x))?

f(g(x)) is represented by 2x² - 2.

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