Function Operations And Domains Exploring F(x) = X^2 - 7x And G(x) = 6 + X

In mathematics, functions are fundamental building blocks, and understanding how to manipulate and combine them is crucial. This article delves into the world of function operations, focusing on two specific functions: f(x) = x^2 - 7x and g(x) = 6 + x. We'll explore how to add, subtract, multiply, and divide these functions, and we'll also determine the domain of the resulting quotient function. By working through these examples, you'll gain a deeper understanding of function composition and the importance of considering domain restrictions.

(a) Finding (f + g)(x): Adding Functions

The first operation we'll explore is the addition of two functions. When we add functions, we simply add their corresponding outputs for the same input value. In other words, (f + g)(x) is found by adding f(x) and g(x). This process involves combining like terms and simplifying the expression. Understanding function addition is a foundational concept in calculus and other advanced mathematical topics, as it allows us to build more complex functions from simpler ones. Moreover, the domain of the resulting function is the intersection of the domains of the original functions, which is an important consideration in many applications.

To find (f + g)(x), we begin by writing out the expressions for f(x) and g(x): f(x) = x^2 - 7x and g(x) = 6 + x. Now, we add these two expressions together:

(f + g)(x) = f(x) + g(x) = (x^2 - 7x) + (6 + x).

Next, we combine like terms. We have a quadratic term (x^2), two linear terms (-7x and x), and a constant term (6). Combining the linear terms, we get -7x + x = -6x. Therefore,

(f + g)(x) = x^2 - 6x + 6.

This resulting quadratic function represents the sum of the original functions f(x) and g(x). The domain of both f(x) and g(x) is all real numbers, as there are no restrictions on the input values. Since the domain of the sum of two functions is the intersection of their individual domains, the domain of (f + g)(x) is also all real numbers.

In summary, adding functions involves a straightforward algebraic process of combining their expressions. This operation is a cornerstone of function manipulation and is essential for various mathematical applications. The resulting function, in this case, is a quadratic, and its domain encompasses all real numbers, reflecting the unrestricted nature of the original functions.

(b) Finding (f - g)(x): Subtracting Functions

Next, we'll delve into the subtraction of functions, a process analogous to addition but with a crucial difference: the sign of the subtracted function must be distributed carefully. Subtracting functions involves finding the difference between their outputs for the same input value. This operation, denoted as (f - g)(x), is calculated by subtracting g(x) from f(x). Understanding function subtraction is vital in various mathematical contexts, including finding the difference between two rates of change or analyzing the net effect of two competing processes. Like addition, the domain of the resulting function is the intersection of the domains of the original functions.

To determine (f - g)(x), we again start with the expressions for f(x) and g(x): f(x) = x^2 - 7x and g(x) = 6 + x. Now, we subtract g(x) from f(x):

(f - g)(x) = f(x) - g(x) = (x^2 - 7x) - (6 + x).

The key step in subtraction is to distribute the negative sign to all terms within the parentheses of g(x). This gives us:

(f - g)(x) = x^2 - 7x - 6 - x.

Now, we combine like terms. We have the quadratic term (x^2), the linear terms (-7x and -x), and the constant term (-6). Combining the linear terms, we get -7x - x = -8x. Thus,

(f - g)(x) = x^2 - 8x - 6.

This resulting quadratic function represents the difference between f(x) and g(x). Similar to the addition case, the domain of both f(x) and g(x) is all real numbers. Therefore, the domain of (f - g)(x), being the intersection of the individual domains, is also all real numbers.

In essence, subtracting functions requires careful attention to the distribution of the negative sign. This operation provides a valuable tool for analyzing the disparities between functions and is widely used in various mathematical and scientific applications. The resulting function, in this case, is another quadratic with a domain encompassing all real numbers.

(c) Finding (f ⋅ g)(x): Multiplying Functions

Moving on to another fundamental operation, we consider the multiplication of functions. Multiplying functions involves finding the product of their outputs for the same input value. This operation, denoted as (f ⋅ g)(x), is calculated by multiplying f(x) and g(x). Understanding function multiplication is crucial in various mathematical contexts, such as finding the area under a curve or modeling the combined effect of two factors. Like addition and subtraction, the domain of the resulting function is generally the intersection of the domains of the original functions.

To calculate (f ⋅ g)(x), we start with the expressions for f(x) and g(x): f(x) = x^2 - 7x and g(x) = 6 + x. We then multiply these two expressions together:

(f ⋅ g)(x) = f(x) ⋅ g(x) = (x^2 - 7x)(6 + x).

To multiply these expressions, we use the distributive property (often referred to as FOIL): multiply each term in the first expression by each term in the second expression.

(f ⋅ g)(x) = x^2(6 + x) - 7x(6 + x) = 6x^2 + x^3 - 42x - 7x^2.

Now, we combine like terms. We have a cubic term (x^3), quadratic terms (6x^2 and -7x^2), and a linear term (-42x). Combining the quadratic terms, we get 6x^2 - 7x^2 = -x^2. Therefore,

(f ⋅ g)(x) = x^3 - x^2 - 42x.

This resulting cubic function represents the product of f(x) and g(x). Since the domain of both f(x) and g(x) is all real numbers, the domain of (f ⋅ g)(x), the intersection of their domains, is also all real numbers.

In essence, multiplying functions involves distributing and combining like terms. This operation is essential for modeling situations where the output depends on the product of two factors. The resulting function, in this case, is a cubic, and its domain encompasses all real numbers.

(d) Finding (f / g)(x): Dividing Functions

The division of functions introduces a new layer of complexity, as we must consider potential restrictions on the domain. Dividing functions involves finding the quotient of their outputs for the same input value. This operation, denoted as (f / g)(x), is calculated by dividing f(x) by g(x). Understanding function division is crucial in various mathematical contexts, such as finding the ratio of two quantities or analyzing rates of change. However, a critical consideration arises: we cannot divide by zero. Therefore, the domain of the resulting function is the intersection of the domains of the original functions, excluding any values where the denominator function, g(x), equals zero.

To determine (f / g)(x), we begin with the expressions for f(x) and g(x): f(x) = x^2 - 7x and g(x) = 6 + x. We then divide f(x) by g(x):

(f / g)(x) = f(x) / g(x) = (x^2 - 7x) / (6 + x).

This rational function represents the quotient of f(x) and g(x). To simplify, we can factor the numerator:

(f / g)(x) = x(x - 7) / (6 + x).

This simplified form highlights the potential for domain restrictions. The denominator, 6 + x, cannot be equal to zero. This leads us to the next crucial step: determining the domain of the quotient function.

(e) The Domain of (f / g)(x):

As mentioned earlier, the domain of a quotient function is the intersection of the domains of the original functions, excluding any values where the denominator is zero. In this case, the domains of both f(x) and g(x) are all real numbers. However, we must exclude any values of x that make the denominator, g(x) = 6 + x, equal to zero.

To find these values, we set the denominator equal to zero and solve for x:

6 + x = 0

x = -6

Therefore, x = -6 is the value that makes the denominator zero, and we must exclude it from the domain. The domain of (f / g)(x) is all real numbers except for x = -6. We can express this in interval notation as:

Domain of (f / g)(x) = (-∞, -6) ∪ (-6, ∞).

This domain reflects the crucial restriction that division by zero is undefined. The quotient function is defined for all real numbers except for x = -6, where the denominator becomes zero.

In conclusion, dividing functions introduces the important consideration of domain restrictions. The resulting rational function has a domain that excludes any values that make the denominator zero. This understanding is critical for accurately analyzing and interpreting function behavior.

Summary

In this exploration, we've dissected the operations of addition, subtraction, multiplication, and division applied to functions. We've seen how these operations combine functions to create new ones, each with its unique properties and behaviors. The functions f(x) = x^2 - 7x and g(x) = 6 + x served as our examples, allowing us to apply the concepts concretely. We've emphasized the importance of considering the domain when working with function operations, particularly in the case of division, where the denominator cannot be zero. These fundamental concepts are vital for further studies in calculus and other advanced mathematical fields, as they provide the groundwork for understanding more complex function relationships and applications.