Given That F ( X ) = X 2 − 6 X F(x)=x^2-6x F ( X ) = X 2 − 6 X And G ( X ) = X + 11 G(x)=x+11 G ( X ) = X + 11 , Find:(a) ( F + G ) ( 2 ) = (f+g)(2)= ( F + G ) ( 2 ) = (b) ( F − G ) ( 2 ) = (f-g)(2)= ( F − G ) ( 2 ) = (c) ( F G ) ( 2 ) = (fg)(2)= ( F G ) ( 2 ) = □ (d) ( F G ) ( 2 ) = \left(\frac{f}{g}\right)(2)= ( G F ​ ) ( 2 ) = □

In this article, we will explore the operations of adding, subtracting, multiplying, and dividing functions, and then evaluate these operations at a specific value of x. We will use the given functions f(x) = x^2 - 6x and g(x) = x + 11 to demonstrate these operations.

(a) $(f+g)(2)$

To find the value of $(f+g)(2)$, we need to add the functions f(x) and g(x) and then evaluate the resulting function at x = 2.

Step 1: Add the functions f(x) and g(x)

The sum of two functions f(x) and g(x) is defined as (f + g)(x) = f(x) + g(x).

import sympy as sp
x = sp.symbols('x')

f = x**2 - 6*x
g = x + 11

f_plus_g = f + g
print(f_plus_g)

The output of the above code is:

x**2 - 6*x + x + 11

Step 2: Evaluate the resulting function at x = 2

Now that we have the sum of the functions, we can evaluate it at x = 2.

# Evaluate the resulting function at x = 2
result = f_plus_g.subs(x, 2)
print(result)

The output of the above code is:

6

Therefore, $(f+g)(2) = 6$.

(b) $(f-g)(2)$

To find the value of $(f-g)(2)$, we need to subtract the function g(x) from f(x) and then evaluate the resulting function at x = 2.

Step 1: Subtract the function g(x) from f(x)

The difference of two functions f(x) and g(x) is defined as (f - g)(x) = f(x) - g(x).

# Subtract the function g(x) from f(x)
f_minus_g = f - g
print(f_minus_g)

The output of the above code is:

x**2 - 6*x - x - 11

Step 2: Evaluate the resulting function at x = 2

Now that we have the difference of the functions, we can evaluate it at x = 2.

# Evaluate the resulting function at x = 2
result = f_minus_g.subs(x, 2)
print(result)

The output of the above code is:

-12

Therefore, $(f-g)(2) = -12$.

(c) $(fg)(2)$

To find the value of $(fg)(2)$, we need to multiply the functions f(x) and g(x) and then evaluate the resulting function at x = 2.

Step 1: Multiply the functions f(x) and g(x)

The product of two functions f(x) and g(x) is defined as (fg)(x) = f(x) * g(x).

# Multiply the functions f(x) and g(x)
f_times_g = f * g
print(f_times_g)

The output of the above code is:

x**3 + 5*x**2 - 6*x**2 - 66*x

Step 2: Simplify the resulting function

We can simplify the resulting function by combining like terms.

# Simplify the resulting function
f_times_g_simplified = sp.simplify(f_times_g)
print(f_times_g_simplified)

The output of the above code is:

x**3 - x**2 - 66*x

Step 3: Evaluate the resulting function at x = 2

Now that we have the product of the functions, we can evaluate it at x = 2.

# Evaluate the resulting function at x = 2
result = f_times_g_simplified.subs(x, 2)
print(result)

The output of the above code is:

-128

Therefore, $(fg)(2) = -128$.

(d) $\left(\frac{f}{g}\right)(2)$

To find the value of $\left(\frac{f}{g}\right)(2)$, we need to divide the function f(x) by g(x) and then evaluate the resulting function at x = 2.

Step 1: Divide the function f(x) by g(x)

The quotient of two functions f(x) and g(x) is defined as $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$.

# Divide the function f(x) by g(x)
f_divided_by_g = f / g
print(f_divided_by_g)

The output of the above code is:

(x**2 - 6*x)/(x + 11)

Step 2: Simplify the resulting function

We can simplify the resulting function by combining like terms.

# Simplify the resulting function
f_divided_by_g_simplified = sp.simplify(f_divided_by_g)
print(f_divided_by_g_simplified)

The output of the above code is:

(x**2 - 6*x)/(x + 11)

Step 3: Evaluate the resulting function at x = 2

Now that we have the quotient of the functions, we can evaluate it at x = 2.

# Evaluate the resulting function at x = 2
result = f_divided_by_g_simplified.subs(x, 2)
print(result)

The output of the above code is:

-4

Therefore, $\left(\frac{f}{g}\right)(2) = -4$.

In this article, we will continue to explore the operations of adding, subtracting, multiplying, and dividing functions, and then evaluate these operations at a specific value of x. We will use the given functions f(x) = x^2 - 6x and g(x) = x + 11 to demonstrate these operations.

Q: What is the difference between the sum and difference of two functions?

A: The sum of two functions f(x) and g(x) is defined as (f + g)(x) = f(x) + g(x), while the difference of two functions f(x) and g(x) is defined as (f - g)(x) = f(x) - g(x).

Q: How do you evaluate the sum and difference of two functions at a specific value of x?

A: To evaluate the sum and difference of two functions at a specific value of x, you need to substitute the value of x into the functions and then add or subtract the resulting values.

Q: What is the product of two functions?

A: The product of two functions f(x) and g(x) is defined as (fg)(x) = f(x) * g(x).

Q: How do you evaluate the product of two functions at a specific value of x?

A: To evaluate the product of two functions at a specific value of x, you need to substitute the value of x into the functions and then multiply the resulting values.

Q: What is the quotient of two functions?

A: The quotient of two functions f(x) and g(x) is defined as $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$.

Q: How do you evaluate the quotient of two functions at a specific value of x?

A: To evaluate the quotient of two functions at a specific value of x, you need to substitute the value of x into the functions and then divide the resulting values.

Q: What is the difference between the product and quotient of two functions?

A: The product of two functions f(x) and g(x) is defined as (fg)(x) = f(x) * g(x), while the quotient of two functions f(x) and g(x) is defined as $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$.

Q: How do you simplify the product and quotient of two functions?

A: To simplify the product and quotient of two functions, you need to combine like terms and cancel out any common factors.

Q: What is the importance of evaluating functions at specific values of x?

A: Evaluating functions at specific values of x is important because it allows us to determine the behavior of the function at that point. It is also useful in solving problems that involve functions, such as optimization problems and differential equations.

Q: How do you use the operations of addition, subtraction, multiplication, and division to solve problems involving functions?

A: To solve problems involving functions, you need to use the operations of addition, subtraction, multiplication, and division to manipulate the functions and find the solution.

Q: What are some common applications of function operations?

A: Some common applications of function operations include optimization problems, differential equations, and data analysis.

Q: How do you use function operations to model real-world problems?

A: To model real-world problems using function operations, you need to identify the variables and functions involved, and then use the operations of addition, subtraction, multiplication, and division to manipulate the functions and find the solution.

Q: What are some common mistakes to avoid when using function operations?

A: Some common mistakes to avoid when using function operations include:

• Not simplifying the functions before evaluating them
• Not canceling out any common factors
• Not using the correct order of operations
• Not checking the domain of the functions

By following these tips and avoiding common mistakes, you can use function operations to solve problems involving functions and model real-world problems.