Function Operations With F(x) = X^2 - 7x And G(x) = 6 + X
In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding relationships between variables. Operations on functions, such as addition, subtraction, multiplication, and division, allow us to combine and manipulate these relationships in powerful ways. In this article, we delve into the world of function operations, using the specific examples of f(x) = x^2 - 7x and g(x) = 6 + x to illustrate the concepts. We will explore how to perform these operations, determine the resulting functions, and identify any restrictions on their domains.
(a) Finding (f + g)(x)
The sum of two functions, denoted as (f + g)(x), is obtained by simply adding the expressions for the individual functions. In other words, we combine like terms from f(x) and g(x) to arrive at the new function.
Given our functions, f(x) = x^2 - 7x and g(x) = 6 + x, we can find (f + g)(x) as follows:
(f + g)(x) = f(x) + g(x)
Substitute the expressions for f(x) and g(x):
(f + g)(x) = (x^2 - 7x) + (6 + x)
Now, combine like terms:
(f + g)(x) = x^2 - 7x + x + 6
(f + g)(x) = x^2 - 6x + 6
Therefore, the sum of the functions f(x) and g(x) is the quadratic function (f + g)(x) = x^2 - 6x + 6. This new function represents the combined behavior of the original two functions.
This process of adding functions is straightforward and allows us to create new functions that capture the combined effects of the original ones. The resulting function, in this case, is a quadratic, which exhibits a parabolic shape when graphed.
(b) Finding (f - g)(x)
Similar to addition, the difference of two functions, denoted as (f - g)(x), is found by subtracting the expression for g(x) from the expression for f(x). It's crucial to pay attention to the order of subtraction, as f - g is generally not the same as g - f.
Using our functions, f(x) = x^2 - 7x and g(x) = 6 + x, we calculate (f - g)(x) as follows:
(f - g)(x) = f(x) - g(x)
Substitute the expressions for f(x) and g(x):
(f - g)(x) = (x^2 - 7x) - (6 + x)
Distribute the negative sign to all terms in g(x):
(f - g)(x) = x^2 - 7x - 6 - x
Combine like terms:
(f - g)(x) = x^2 - 8x - 6
Thus, the difference between the functions f(x) and g(x) is the quadratic function (f - g)(x) = x^2 - 8x - 6. This function represents the result of subtracting the values of g(x) from the values of f(x) for each x.
The subtraction of functions highlights how the relative values of the functions change as x varies. The resulting quadratic function provides a distinct representation of this relationship.
(c) Finding (f ⋅ g)(x)
The product of two functions, denoted as (f ⋅ g)(x), is obtained by multiplying the expressions for f(x) and g(x). This involves distributing each term in f(x) over each term in g(x).
For our functions, f(x) = x^2 - 7x and g(x) = 6 + x, we determine (f ⋅ g)(x) as follows:
(f ⋅ g)(x) = f(x) ⋅ g(x)
Substitute the expressions for f(x) and g(x):
(f ⋅ g)(x) = (x^2 - 7x) ⋅ (6 + x)
Use the distributive property (also known as the FOIL method) to multiply the expressions:
(f ⋅ g)(x) = x^2 ⋅ (6 + x) - 7x ⋅ (6 + x)
(f ⋅ g)(x) = 6x^2 + x^3 - 42x - 7x^2
Combine like terms:
(f ⋅ g)(x) = x^3 - x^2 - 42x
Therefore, the product of the functions f(x) and g(x) is the cubic function (f ⋅ g)(x) = x^3 - x^2 - 42x. This function represents the result of multiplying the values of f(x) and g(x) for each x.
The multiplication of functions can lead to functions with higher degrees, as seen in this case where a cubic function is produced from a quadratic and a linear function. The resulting function exhibits a more complex behavior than the original functions.
(d) Finding (f/g)(x)
The quotient of two functions, denoted as (f/g)(x), is defined as the ratio of f(x) to g(x), provided that g(x) is not equal to zero. This operation introduces the possibility of restrictions on the domain, as we cannot divide by zero.
Given our functions, f(x) = x^2 - 7x and g(x) = 6 + x, we find (f/g)(x) as follows:
(f/g)(x) = f(x) / g(x)
Substitute the expressions for f(x) and g(x):
(f/g)(x) = (x^2 - 7x) / (6 + x)
We can factor the numerator to see if any simplification is possible:
(f/g)(x) = x(x - 7) / (6 + x)
In this case, there are no common factors between the numerator and denominator that can be canceled. Therefore, the quotient of the functions is:
(f/g)(x) = x(x - 7) / (6 + x)
This rational function represents the division of the values of f(x) by the values of g(x), with the crucial caveat that the denominator cannot be zero.
(e) Determining the Domain of (f/g)(x)
The domain of a function is the set of all possible input values (x-values) for which the function is defined. When dealing with quotients of functions, we must exclude any values of x that make the denominator zero, as division by zero is undefined.
For the quotient function (f/g)(x) = x(x - 7) / (6 + x), the denominator is (6 + x). To find the values of x that make the denominator zero, we set the denominator equal to zero and solve for x:
6 + x = 0
x = -6
Therefore, x = -6 is the only value that makes the denominator zero. This value must be excluded from the domain of (f/g)(x).
The domain of (f/g)(x) consists of all real numbers except for x = -6. We can express this in several ways:
- Set notation: {x | x ∈ ℝ, x ≠ -6}
- Interval notation: (-∞, -6) ∪ (-6, ∞)
In interval notation, we use parentheses to indicate that -6 is not included in the domain. The union symbol (∪) combines the two intervals representing all real numbers less than -6 and all real numbers greater than -6.
Understanding the domain of a function is crucial for interpreting its behavior and avoiding undefined values. In the case of quotients, identifying and excluding values that make the denominator zero is a critical step.
In this article, we have explored the fundamental operations on functions – addition, subtraction, multiplication, and division – using the examples of f(x) = x^2 - 7x and g(x) = 6 + x. We have demonstrated how to perform these operations, determine the resulting functions, and identify restrictions on the domain, particularly in the case of quotients. By mastering these operations, we gain a powerful toolkit for manipulating and understanding relationships between variables, paving the way for more advanced mathematical concepts and applications.
The ability to combine functions through these operations allows us to model complex phenomena by building upon simpler functional relationships. Understanding the domains of the resulting functions ensures that our models are valid and provide meaningful insights.
