Math Problem Statement

The image contains a math problem involving functions. The tasks include performing operations such as addition, subtraction, and multiplication on the given functions f(x) = -x^2 + 3x + 6 and g(x) = 3x^2 - 5x.

Solution

The image contains a math problem involving functions. Here’s the breakdown of what is visible on the board:

Given functions:

  1. f(x)=−x2+3x+6f(x) = -x^2 + 3x + 6
  2. g(x)=3x2−5xg(x) = 3x^2 - 5x
  3. h(x)=6x+1h(x) = 6x + 1

Tasks: a. Calculate f(x)+g(x)f(x) + g(x). b. Calculate (f−g)(x)(f - g)(x). c. Calculate (f×g)(x)(f \times g)(x).

Let's go through each task:

a. f(x)+g(x)f(x) + g(x):

We add the two functions: f(x)+g(x)=(−x2+3x+6)+(3x2−5x)f(x) + g(x) = (-x^2 + 3x + 6) + (3x^2 - 5x) Simplify:

= 2x^2 - 2x + 6$$ ### b. $$(f - g)(x)$$: We subtract $$g(x)$$ from $$f(x)$$: $$(f - g)(x) = (-x^2 + 3x + 6) - (3x^2 - 5x)$$ Simplify: $$= -x^2 - 3x^2 + 3x + 5x + 6 = -4x^2 + 8x + 6$$ ### c. $$(f \times g)(x)$$: We multiply $$f(x)$$ and $$g(x)$$: $$(f \times g)(x) = (-x^2 + 3x + 6)(3x^2 - 5x)$$ Using distribution (FOIL): $$= (-x^2)(3x^2) + (-x^2)(-5x) + (3x)(3x^2) + (3x)(-5x) + (6)(3x^2) + (6)(-5x)$$ Simplifying each term: $$= -3x^4 + 5x^3 + 9x^3 - 15x^2 + 18x^2 - 30x$$ Combine like terms: $$= -3x^4 + 14x^3 + 3x^2 - 30x$$ Would you like further explanation or any clarifications? ### Related Questions: 1. How would you solve for $$f(x) + h(x)$$? 2. Can you evaluate $$(g \times h)(x)$$? 3. What is the result of dividing $$f(x)$$ by $$g(x)$$? 4. How would you find the roots of $$f(x) + g(x)$$? 5. Can you find the derivative of $$f(x)$$? **Tip**: Always combine like terms carefully in polynomials to avoid mistakes during simplification.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Functions
Operations on Functions

Formulas

Addition of functions: (f + g)(x) = f(x) + g(x)
Subtraction of functions: (f - g)(x) = f(x) - g(x)
Multiplication of functions: (f × g)(x) = f(x) × g(x)

Theorems

-

Suitable Grade Level

Grades 9-11

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