College

Given functions [tex]f[/tex] and [tex]g[/tex], perform the indicated operations.

[tex]f(x) = 3x - 6[/tex]; [tex]g(x) = 5x + 4[/tex]

Find [tex]f \cdot g[/tex].

A. [tex]15x^2 - 18x - 24[/tex]

B. [tex]8x^2 - 18x - 2[/tex]

C. [tex]15x^2 - 24[/tex]

D. [tex]15x^2 - 26x - 24[/tex]

Answer :

To find the product [tex]\( fg(x) \)[/tex] of the functions [tex]\( f(x) = 3x - 6 \)[/tex] and [tex]\( g(x) = 5x + 4 \)[/tex], we'll multiply the two expressions using the distributive property.

Step-by-step solution:

1. Write down the expressions for the functions:
- [tex]\( f(x) = 3x - 6 \)[/tex]
- [tex]\( g(x) = 5x + 4 \)[/tex]

2. Use the distributive property to expand [tex]\((3x - 6) \times (5x + 4)\)[/tex]:

[tex]\[
fg(x) = (3x - 6)(5x + 4)
\][/tex]

We'll distribute each term in the first expression to each term in the second expression.

3. Multiply [tex]\(3x\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
- [tex]\(3x \times 5x = 15x^2\)[/tex]
- [tex]\(3x \times 4 = 12x\)[/tex]

4. Multiply [tex]\(-6\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
- [tex]\(-6 \times 5x = -30x\)[/tex]
- [tex]\(-6 \times 4 = -24\)[/tex]

5. Combine all the results:
[tex]\[
fg(x) = 15x^2 + 12x - 30x - 24
\][/tex]

6. Combine the like terms ([tex]\(12x\)[/tex] and [tex]\(-30x\)[/tex]):
[tex]\[
fg(x) = 15x^2 - 18x - 24
\][/tex]

So, the result of the function multiplication [tex]\( fg(x) \)[/tex] is [tex]\( 15x^2 - 18x - 24 \)[/tex].

Therefore, the correct choice is:

A. [tex]\( 15x^2 - 18x - 24 \)[/tex]

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