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Let \( f(x) = 5x + 2 \) and \( g(x) = 4x^2 + 5x \).

Match the statements defined below with the letters labeling their equivalent expressions. You must get all of the answers correct to receive credit.

1. \( g \circ g \)
2. \( f \circ f \)
3. \( f \circ g \)
4. \( g \circ f \)

A. \( 20x^2 + 25x + 2 \)
B. \( 25x + 12 \)
C. \( 64x^4 + 160x^3 + 120x^2 + 25x \)
D. \( 100x^2 + 105x + 26 \)

Note: You can earn partial credit on this problem.

Answer :

Final answer:

The compositions of the functions f(x) and g(x) are evaluated by substituting one function into the other and then simplifying. The result for gºg is C, fºf gives B, fog gives A, and gof concludes to D.

Explanation:

The concept at play here is function composition, where you substitute one function into another. Here's how you find each of the required expressions:

  1. gºg: You plug g(x) into itself, so you would substitute 4x² + 5x into g(x) = 4x² + 5x to get g(g(x)) = 4(4x² + 5x)² + 5(4x² + 5x), which simplifies to C. 64x4 + 160x3 + 120x2 + 25x.
  2. fºf: Similarly, substituting f(x) into itself gives f(f(x)) = 5(5x + 2) + 2, which simplifies to B. 25x + 12.
  3. fog: Substituting g(x) into f(x) gives f(g(x)) = 5(4x² + 5x) + 2, which equates to A. 20x2 + 25x + 2.
  4. gof: Finally, substituting f(x) into g(x) results in g(f(x)) = 4(5x + 2)² + 5(5x + 2), which simplifies to D. 100x2 + 105x + 26.

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