Math Problem Statement

Solve the following operations. Given: f(x) = 2x + 3, g(x) = 3x - 2, h(x) = 6x^2 + 5x - 6. 1. (f+g)(x) 2. (f+g)(-2) 3. (f ⋅ g)(x) 4. (f ⋅ g)(2) 5. (f ⋅ g)(x) 6. (f ⋅ g)(-2) 7. f/h(x) 8. f/h(-1)

Solution

The image contains a list of given functions and several operations to solve. The functions are:

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

The tasks are numbered, so let's go through them one by one:

1. (f+g)(x)(f+g)(x)

This means adding the functions f(x)f(x) and g(x)g(x): (f+g)(x)=f(x)+g(x)=(2x+3)+(3x−2)=5x+1(f + g)(x) = f(x) + g(x) = (2x + 3) + (3x - 2) = 5x + 1

2. (f+g)(−2)(f+g)(-2)

This means evaluating (f+g)(x)(f + g)(x) at x=−2x = -2: (f+g)(−2)=5(−2)+1=−10+1=−9(f + g)(-2) = 5(-2) + 1 = -10 + 1 = -9

3. (f⋅g)(x)(f \cdot g)(x)

This is the product of f(x)f(x) and g(x)g(x): (f⋅g)(x)=f(x)⋅g(x)=(2x+3)(3x−2)(f \cdot g)(x) = f(x) \cdot g(x) = (2x + 3)(3x - 2) Expanding this product:

= 6x^2 - 4x + 9x - 6 = 6x^2 + 5x - 6$$ ### 4. $$(f \cdot g)(2)$$ This means evaluating $$(f \cdot g)(x)$$ at $$x = 2$$: $$(f \cdot g)(2) = 6(2)^2 + 5(2) - 6 = 6(4) + 10 - 6 = 24 + 10 - 6 = 28$$ ### 5. $$(f \cdot g)(x)$$ (same as Task 3) We already solved this in Task 3: $$(f \cdot g)(x) = 6x^2 + 5x - 6$$ ### 6. $$(f \cdot g)(-2)$$ This means evaluating $$(f \cdot g)(x)$$ at $$x = -2$$: $$(f \cdot g)(-2) = 6(-2)^2 + 5(-2) - 6 = 6(4) - 10 - 6 = 24 - 10 - 6 = 8$$ ### 7. $$\frac{f}{h}(x)$$ This is the ratio $$\frac{f(x)}{h(x)}$$: $$\frac{f}{h}(x) = \frac{2x + 3}{6x^2 + 5x - 6}$$ ### 8. $$\frac{f}{h}(-1)$$ This means evaluating $$\frac{f}{h}(x)$$ at $$x = -1$$: $$f(-1) = 2(-1) + 3 = -2 + 3 = 1$$ $$h(-1) = 6(-1)^2 + 5(-1) - 6 = 6(1) - 5 - 6 = 6 - 5 - 6 = -5$$ Thus: $$\frac{f}{h}(-1) = \frac{1}{-5} = -\frac{1}{5}$$ --- Would you like more details on any specific step, or help with another concept? Here are five related questions: 1. How do you simplify the sum of two functions? 2. What are the steps to expand the product of two binomials? 3. How can you calculate the quotient of two functions? 4. What happens when you evaluate composite functions? 5. How can function evaluation at specific points help in graphing functions? **Tip:** Always keep track of parentheses when performing operations with functions to avoid errors!

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Operations with Functions
Evaluating Functions

Formulas

Sum of Functions: (f + g)(x) = f(x) + g(x)
Product of Functions: (f ⋅ g)(x) = f(x) ⋅ g(x)
Quotient of Functions: (f/h)(x) = f(x)/h(x)

Theorems

Basic Function Operations

Suitable Grade Level

Grades 9-10

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