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Use the functions below.

f(x)=x2−9
g(x)=x2−x−12
What is (fg)(x)
?

Enter the missing values.

(1 point)
(fg)(x)=
$$

1 answer

To find \((fg)(x)\), we need to multiply the functions \(f(x)\) and \(g(x)\).

Given:

  • \(f(x) = x^2 - 9\)
  • \(g(x) = x^2 - x - 12\)

Now, we calculate \((fg)(x)\):

\[ (fg)(x) = f(x) \cdot g(x) = (x^2 - 9)(x^2 - x - 12) \]

To perform the multiplication, we will use the distributive property (also known as the FOIL method for binomials):

\[ (fg)(x) = (x^2)(x^2) + (x^2)(-x) + (x^2)(-12) + (-9)(x^2) + (-9)(-x) + (-9)(-12) \]

Calculating each term:

  1. \(x^2 \cdot x^2 = x^4\)
  2. \(x^2 \cdot -x = -x^3\)
  3. \(x^2 \cdot -12 = -12x^2\)
  4. \(-9 \cdot x^2 = -9x^2\)
  5. \(-9 \cdot -x = 9x\)
  6. \(-9 \cdot -12 = 108\)

Now, combine all the terms:

\[ (fg)(x) = x^4 - x^3 - 12x^2 - 9x^2 + 9x + 108 \]

Combine like terms:

\[ = x^4 - x^3 - 21x^2 + 9x + 108 \]

So the final answer is:

\[ (fg)(x) = x^4 - x^3 - 21x^2 + 9x + 108 \]

Enter the missing values:

\[ (fg)(x) = x^4 - x^3 - 21x^2 + 9x + 108 \]