Function Operations And Domain Calculation For F(x) = X^2 - 7x And G(x) = 6 + X

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This comprehensive guide delves into the fundamental operations involving functions, providing a step-by-step solution to the problem: Given $f(x) = x^2 - 7x$ and $g(x) = 6 + x$, we aim to determine $(f+g)(x)$, $(f-g)(x)$, (f ullet g)(x), $\left(\frac{f}{g}\right)(x)$, and the domain of $\frac{f}{g}$. Understanding these operations is crucial for mastering advanced mathematical concepts and their applications in various fields.

(a) Finding $(f+g)(x)$: Combining Functions Through Addition

To find the sum of two functions, denoted as $(f+g)(x)$, we simply add the expressions of the individual functions together. This operation combines the outputs of the functions for a given input x. In this specific case, we have $f(x) = x^2 - 7x$ and $g(x) = 6 + x$. Therefore, to determine $(f+g)(x)$, we perform the following addition:

$(f+g)(x) = f(x) + g(x)$

Substituting the given expressions for $f(x)$ and $g(x)$, we get:

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

Now, we combine like terms to simplify the expression. Like terms are those that have the same variable raised to the same power. In this case, we have terms with $x^2$, terms with $x$, and constant terms. Combining the x terms, we have -7x + x, which simplifies to -6x. The constant term is simply 6. Thus, we can rewrite the expression as:

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

Therefore, the sum of the functions $f(x)$ and $g(x)$, represented as $(f+g)(x)$, is equal to $x^2 - 6x + 6$. This new function represents the combined output of the original functions for any given input x. Understanding function addition is essential for modeling situations where quantities are combined, such as calculating total cost or total revenue.

(b) Determining $(f-g)(x)$: Function Subtraction

The difference of two functions, represented as $(f-g)(x)$, is found by subtracting the expression of the second function, g(x), from the expression of the first function, f(x). This operation highlights the difference in the outputs of the two functions for the same input x. Given our functions $f(x) = x^2 - 7x$ and $g(x) = 6 + x$, we calculate $(f-g)(x)$ as follows:

$(f-g)(x) = f(x) - g(x)$

Substituting the given expressions, we have:

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

It is crucial to distribute the negative sign correctly when subtracting expressions. The negative sign in front of the parentheses applies to every term inside the parentheses. Therefore, we rewrite the expression as:

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

Now, we combine like terms, similar to the addition operation. We have a term with $x^2$, terms with x, and a constant term. Combining the x terms, we have -7x - x, which simplifies to -8x. The constant term is -6. Thus, the expression becomes:

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

Therefore, the difference of the functions $f(x)$ and $g(x)$, expressed as $(f-g)(x)$, is $x^2 - 8x - 6$. Function subtraction is useful in scenarios where we need to find the net change or difference between two quantities, such as profit calculation (revenue minus cost) or determining the difference in population sizes.

(c) Calculating (f ullet g)(x): Function Multiplication

The product of two functions, denoted as (f ullet g)(x), is obtained by multiplying the expressions of the individual functions. This operation shows how the outputs of the two functions interact multiplicatively for a given input x. With $f(x) = x^2 - 7x$ and $g(x) = 6 + x$, we find (f ullet g)(x) by performing the multiplication:

(f ullet g)(x) = f(x) ullet g(x)

Substituting the expressions for $f(x)$ and $g(x)$, we get:

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

To multiply these expressions, we use the distributive property (often referred to as the FOIL method for binomials). This involves multiplying each term in the first expression by each term in the second expression. We multiply $x^2$ by both 6 and x, and then multiply -7x by both 6 and x:

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

Now, we perform the multiplications:

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

Finally, we combine like terms to simplify the expression. We have terms with $x^3$, terms with $x^2$, and terms with x. Combining the $x^2$ terms, we have 6x^2 - 7x^2, which simplifies to -x^2. Thus, the expression becomes:

(f ullet g)(x) = x^3 - x^2 - 42x

Therefore, the product of the functions $f(x)$ and $g(x)$, represented as (f ullet g)(x), is $x^3 - x^2 - 42x$. Function multiplication is useful in scenarios where quantities are related multiplicatively, such as calculating the area of a rectangle with sides defined by functions or modeling exponential growth.

(d) Evaluating $\left(\frac{f}{g}\right)(x)$: Function Division

The quotient of two functions, represented as $\left(\frac{f}{g}\right)(x)$, is obtained by dividing the expression of the first function, f(x), by the expression of the second function, g(x). This operation highlights the ratio of the outputs of the two functions for a given input x. However, it's crucial to note that this operation is only valid when the denominator, g(x), is not equal to zero. Given our functions $f(x) = x^2 - 7x$ and $g(x) = 6 + x$, we find $\left(\frac{f}{g}\right)(x)$ as follows:

$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$

Substituting the given expressions, we have:

$\left(\frac{f}{g}\right)(x) = \frac{x^2 - 7x}{6 + x}$

Now, we can try to simplify the expression by factoring the numerator. We can factor out an x from the expression $x^2 - 7x$:

$x^2 - 7x = x(x - 7)$

Therefore, we can rewrite the quotient as:

$\left(\frac{f}{g}\right)(x) = \frac{x(x - 7)}{6 + x}$

In this case, we cannot simplify the expression further because there are no common factors between the numerator and the denominator. Thus, the quotient of the functions $f(x)$ and $g(x)$, expressed as $\left(\frac{f}{g}\right)(x)$, is $\frac{x(x - 7)}{6 + x}$. Function division is essential in situations where we need to find a ratio or rate of change, such as calculating average speed or concentration.

(e) Determining the Domain of $\frac{f}{g}$: Identifying Restrictions

The domain of a function is the set of all possible input values (x-values) for which the function produces a valid output. When dealing with the quotient of two functions, $\frac{f}{g}$, the domain is restricted by the fact that we cannot divide by zero. Therefore, the domain of $\frac{f}{g}$ consists of all real numbers x such that g(x) is not equal to zero.

In our case, we have $g(x) = 6 + x$. To find the values of x that make g(x) equal to zero, we set the expression equal to zero and solve for x:

$6 + x = 0$

Subtracting 6 from both sides, we get:

$x = -6$

This means that g(x) is equal to zero when x is -6. Therefore, x cannot be -6 in the domain of $\frac{f}{g}$.

In addition to the restriction imposed by the denominator, we should also consider any restrictions on the domains of the original functions, f(x) and g(x). However, in this case, both $f(x) = x^2 - 7x$ and $g(x) = 6 + x$ are polynomials, and polynomials have a domain of all real numbers. Therefore, the only restriction on the domain of $\frac{f}{g}$ comes from the denominator.

Thus, the domain of $\frac{f}{g}$ is all real numbers x except for x = -6. We can express this in interval notation as:

$(-\infty, -6) \cup (-6, \infty)$

This notation indicates that the domain includes all numbers less than -6 and all numbers greater than -6, but not -6 itself. Understanding the domain of a function is critical for interpreting its behavior and ensuring that the function produces meaningful results. Identifying restrictions such as division by zero is a fundamental aspect of function analysis.

By working through this problem, we have gained a solid understanding of how to perform basic operations on functions, including addition, subtraction, multiplication, and division. We have also learned how to determine the domain of a function, paying particular attention to restrictions imposed by division by zero. These skills are essential for success in calculus and other advanced mathematical topics.