Functions F(x) And G(x) A Comprehensive Exploration
This article delves into the world of functions, specifically focusing on two linear functions, f(x) = 2x + 9 and g(x) = 9x - 4. We will explore various operations that can be performed on these functions, including addition, subtraction, multiplication, and division. Additionally, we will meticulously determine the domain of each resulting function. Understanding the domain is crucial as it defines the set of all possible input values (x-values) for which the function produces a valid output. This comprehensive guide will provide a step-by-step explanation of each operation, ensuring clarity and a solid understanding of function manipulation.
(a) Finding (f + g)(x) and its Domain
The first operation we will tackle is the addition of the two functions, denoted as (f + g)(x). This involves combining the expressions for f(x) and g(x). To find (f + g)(x), we simply add the corresponding terms of the two functions:
(f + g)(x) = f(x) + g(x)
Substituting the given functions, we get:
(f + g)(x) = (2x + 9) + (9x - 4)
Now, we combine like terms:
(f + g)(x) = 2x + 9x + 9 - 4
(f + g)(x) = 11x + 5
Therefore, the sum of the functions f(x) and g(x) is (f + g)(x) = 11x + 5. This resulting function is also a linear function.
Next, we need to determine 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. For linear functions, the domain is typically all real numbers, unless there are specific restrictions. In this case, since (f + g)(x) = 11x + 5 is a linear function with no denominators or radicals, there are no restrictions on the input values.
Thus, the domain of (f + g)(x) is all real numbers, which can be expressed in interval notation as (-∞, ∞). In simpler terms, you can plug in any real number for x, and the function will produce a valid output.
(b) Finding (f - g)(x) and its Domain
Now, let's move on to the subtraction of the two functions, denoted as (f - g)(x). This involves subtracting the expression for g(x) from the expression for f(x). It's crucial to pay attention to the signs when subtracting. To find (f - g)(x), we subtract g(x) from f(x):
(f - g)(x) = f(x) - g(x)
Substituting the given functions, we get:
(f - g)(x) = (2x + 9) - (9x - 4)
Distribute the negative sign to the terms inside the parentheses of g(x):
(f - g)(x) = 2x + 9 - 9x + 4
Now, combine like terms:
(f - g)(x) = 2x - 9x + 9 + 4
(f - g)(x) = -7x + 13
Therefore, the difference of the functions f(x) and g(x) is (f - g)(x) = -7x + 13. Again, the resulting function is a linear function.
To determine the domain of (f - g)(x), we apply the same reasoning as before. Since (f - g)(x) = -7x + 13 is a linear function without any denominators or radicals, there are no restrictions on the input values. The domain of this function encompasses all real numbers.
Thus, the domain of (f - g)(x) is all real numbers, which is expressed in interval notation as (-∞, ∞). This means that any real number can be substituted for x, and the function will yield a valid output.
(c) Finding (f * g)(x) and its Domain
Next, we will explore the multiplication of the two functions, denoted as (f * g)(x). This involves multiplying the expressions for f(x) and g(x). To find (f * g)(x), we multiply f(x) by g(x):
(f * g)(x) = f(x) * g(x)
Substituting the given functions, we get:
(f * g)(x) = (2x + 9)(9x - 4)
Now, we use the distributive property (also known as the FOIL method) to expand the product:
(f * g)(x) = (2x)(9x) + (2x)(-4) + (9)(9x) + (9)(-4)
(f * g)(x) = 18x² - 8x + 81x - 36
Combine like terms:
(f * g)(x) = 18x² + 73x - 36
Therefore, the product of the functions f(x) and g(x) is (f * g)(x) = 18x² + 73x - 36. This resulting function is a quadratic function.
To determine the domain of (f * g)(x), we again look for any restrictions on the input values. Since (f * g)(x) = 18x² + 73x - 36 is a polynomial function, specifically a quadratic, it has no denominators or radicals. Therefore, there are no restrictions on the input values.
Thus, the domain of (f * g)(x) is all real numbers, represented in interval notation as (-∞, ∞). Any real number can be used as an input for this function, and it will produce a valid output.
(d) Finding (f / g)(x) and its Domain
Now, let's examine the division of the two functions, denoted as (f / g)(x). This involves dividing the expression for f(x) by the expression for g(x). A crucial consideration when dividing functions is that the denominator cannot be zero. To find (f / g)(x), we divide f(x) by g(x):
(f / g)(x) = f(x) / g(x)
Substituting the given functions, we get:
(f / g)(x) = (2x + 9) / (9x - 4)
Therefore, the quotient of the functions f(x) and g(x) is (f / g)(x) = (2x + 9) / (9x - 4). This resulting function is a rational function, which is a ratio of two polynomials.
To determine the domain of (f / g)(x), we must consider the restriction that the denominator cannot be zero. We need to find the values of x that make the denominator, 9x - 4, equal to zero and exclude them from the domain.
Set the denominator equal to zero and solve for x:
9x - 4 = 0
Add 4 to both sides:
9x = 4
Divide both sides by 9:
x = 4/9
So, x = 4/9 is the value that makes the denominator zero. This value must be excluded from the domain. Therefore, the domain of (f / g)(x) is all real numbers except x = 4/9. In interval notation, this is expressed as (-∞, 4/9) U (4/9, ∞). This means the function is defined for all real numbers less than 4/9 and all real numbers greater than 4/9, but not at 4/9 itself.
(e) Finding (f + g)(4)
To find (f + g)(4), we can use the expression we found in part (a), which is (f + g)(x) = 11x + 5. We simply substitute x = 4 into this expression:
(f + g)(4) = 11(4) + 5
(f + g)(4) = 44 + 5
(f + g)(4) = 49
Therefore, (f + g)(4) = 49.
(f) Finding (f - g)(2)
To find (f - g)(2), we can use the expression we found in part (b), which is (f - g)(x) = -7x + 13. We substitute x = 2 into this expression:
(f - g)(2) = -7(2) + 13
(f - g)(2) = -14 + 13
(f - g)(2) = -1
Therefore, (f - g)(2) = -1.
(g) Finding (f * g)(3)
To find (f * g)(3), we can use the expression we found in part (c), which is (f * g)(x) = 18x² + 73x - 36. We substitute x = 3 into this expression:
(f * g)(3) = 18(3)² + 73(3) - 36
(f * g)(3) = 18(9) + 219 - 36
(f * g)(3) = 162 + 219 - 36
(f * g)(3) = 345
Therefore, (f * g)(3) = 345.
(h) Finding (f / g)(1)
To find (f / g)(1), we can use the expression we found in part (d), which is (f / g)(x) = (2x + 9) / (9x - 4). We substitute x = 1 into this expression:
(f / g)(1) = (2(1) + 9) / (9(1) - 4)
(f / g)(1) = (2 + 9) / (9 - 4)
(f / g)(1) = 11 / 5
Therefore, (f / g)(1) = 11/5.
Conclusion
In this comprehensive guide, we explored various operations on the functions f(x) = 2x + 9 and g(x) = 9x - 4, including addition, subtraction, multiplication, and division. We meticulously calculated the resulting functions and determined their domains. We also evaluated these combined functions at specific x-values. This exercise provides a solid foundation for understanding function manipulation and domain considerations, which are fundamental concepts in mathematics.
