https://file.aiquickdraw.com/mathbot/file/1728051199479632cuils.jpg

The question asks to determine the function \( (f \cdot g)(x) \) given the following functions:

- \( f(x) = x - 2 \)
- \( g(x) = x^2 + 5 \)

The operation \( (f \cdot g)(x) \) refers to the multiplication of the two functions, so we need to multiply \( f(x) \) and \( g(x) \).

\[
(f \cdot g)(x) = f(x) \cdot g(x)
\]
\[
= (x - 2)(x^2 + 5)
\]

Now, apply the distributive property (FOIL method) to expand this expression:

\[
(x - 2)(x^2 + 5) = x(x^2 + 5) - 2(x^2 + 5)
\]
\[
= x^3 + 5x - 2x^2 - 10
\]

So, the resulting function is:

\[
(f \cdot g)(x) = x^3 - 2x^2 + 5x - 10
\]

Now, let's match this with the options given in the image:

- \( x^3 - 2x^2 + 5x - 10 \)

The correct answer is the second option.

Would you like further details or have any questions?

---

Here are some related questions to explore:

1. What is the definition of function composition and how does it differ from function multiplication?
2. How would you solve \( (g \cdot f)(x) \) with the same functions \( f(x) = x - 2 \) and \( g(x) = x^2 + 5 \)?
3. What happens if you try to divide two functions, say \( f(x) \) and \( g(x) \)? What would that expression look like?
4. Can you find the derivative of the product of two functions using the product rule?
5. How can you graph the resulting function \( x^3 - 2x^2 + 5x - 10 \)?

**Tip:** Always double-check your work by expanding expressions carefully and ensuring that all terms are included correctly!

Determine the function (f · g)(x) given: f(x) = x - 2, g(x) = x^2 + 5

Learn how to determine the function (f · g)(x) when f(x) = x - 2 and g(x) = x^2 + 5. This problem focuses on algebraic concepts of function multiplication and polynomial expansion using the distributive property.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation