College

Given the functions [tex]f(x) = x + 5[/tex] and [tex]g(x) = 3x - 7[/tex], find [tex](f \cdot g)(x)[/tex].

A. [tex]3x^2 - 35[/tex]
B. [tex]3x^2 + 35[/tex]
C. [tex]3x^2 + 8x - 35[/tex]
D. [tex]3x^2 + 15x + 35[/tex]

Answer :

To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the given functions [tex]\(f(x) = x + 5\)[/tex] and [tex]\(g(x) = 3x - 7\)[/tex].

Here's how you can do it step-by-step:

1. Write down the functions:
- [tex]\(f(x) = x + 5\)[/tex]
- [tex]\(g(x) = 3x - 7\)[/tex]

2. Set up the multiplication:
We want to calculate [tex]\((f \cdot g)(x)\)[/tex], which means multiplying [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = (x + 5)(3x - 7)
\][/tex]

3. Apply the distributive property (FOIL method):
Multiply each term in the first binomial by each term in the second binomial:
[tex]\[
(x + 5)(3x - 7) = x \cdot 3x + x \cdot (-7) + 5 \cdot 3x + 5 \cdot (-7)
\][/tex]

4. Perform the multiplication:
- [tex]\(x \cdot 3x = 3x^2\)[/tex]
- [tex]\(x \cdot (-7) = -7x\)[/tex]
- [tex]\(5 \cdot 3x = 15x\)[/tex]
- [tex]\(5 \cdot (-7) = -35\)[/tex]

5. Combine the results:
Add all the terms together:
[tex]\[
3x^2 - 7x + 15x - 35
\][/tex]

6. Simplify by combining like terms:
Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-7x + 15x = 8x
\][/tex]

7. Write the final simplified expression:
[tex]\[
3x^2 + 8x - 35
\][/tex]

Thus, the product [tex]\((f \cdot g)(x)\)[/tex] is [tex]\(3x^2 + 8x - 35\)[/tex]. This corresponds to the option [tex]\(3x^2 + 8x - 35\)[/tex].