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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].
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].
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