Function Operations: Sum, Difference, Product, Quotient & Domain

Hey guys! Let's dive into the awesome world of functions and explore how we can combine them using some cool operations. We're going to learn how to add, subtract, multiply, and divide functions, and we'll also figure out the domains of these new, combined functions. Basically, the domain is all the possible x values that we can plug into a function without causing any mathematical mayhem (like dividing by zero). We'll be working with two functions: f(x) = 5x - 6 and g(x) = x - 7. Let's get started, shall we?

Finding f + g (The Sum of Functions)

Alright, let's start with finding f + g. This just means we're going to add the two functions together. It's super simple! We take f(x) and g(x) and add their expressions. So, if f(x) = 5x - 6 and g(x) = x - 7, then f(x) + g(x) = (5x - 6) + (x - 7). Now, let's simplify this expression. We combine like terms: the x terms (5x and x) and the constant terms (-6 and -7). This gives us 5x + x = 6x and -6 - 7 = -13. Therefore, f(x) + g(x) = 6x - 13. That wasn't so bad, right?

Now, about the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's all the x values that we can plug into the function without breaking any mathematical rules (like dividing by zero or taking the square root of a negative number in the real number system). For the function f(x) + g(x) = 6x - 13, there are no restrictions on the x values. We can plug in any real number we want, and we'll get a real number back. This is because it's a linear function (a straight line). There are no fractions, no square roots, and no other operations that could cause problems. Thus, the domain of f + g is all real numbers, which we can write as (-∞, ∞). Pretty straightforward!

To recap: when we find the sum of two functions, f + g, we add their expressions. For f(x) = 5x - 6 and g(x) = x - 7, the sum is 6x - 13. The domain of the resulting function, in this case, is all real numbers because there are no restrictions on the x values. It is important to emphasize the significance of domain in this context because, as we move forward to other operations such as division, the domain will be crucial in ensuring that the resulting functions are properly defined and adhere to mathematical rules. Always remember that the domain gives us all the permissible x-values that will not cause the function to be undefined. Let's move on!

Finding f - g (The Difference of Functions)

Now, let's look at the difference of the functions, f - g. This means we're going to subtract g(x) from f(x). It is very important to pay attention to the order here! The subtraction is f(x) - g(x), and this is completely different from g(x) - f(x). We start with f(x) = 5x - 6 and we are going to subtract g(x) = x - 7. So, we have (5x - 6) - (x - 7). Remember to distribute the negative sign to both terms inside the parentheses of g(x). This means we are subtracting x and subtracting –7, which becomes adding 7. Thus, we get 5x - 6 - x + 7. Combining like terms, we get 5x - x = 4x and -6 + 7 = 1. So, f(x) - g(x) = 4x + 1. Cool, huh?

Let's talk about the domain of f - g. Similar to f + g, the resulting function, 4x + 1, is a linear function. There are no restrictions on the x values we can use. We can plug in any real number and get a real number back. Therefore, the domain of f - g is also all real numbers, or (-∞, ∞). Again, simple as pie!

The takeaway: To find f - g, we subtract the expression of g(x) from f(x). For our functions, f(x) = 5x - 6 and g(x) = x - 7, the difference is 4x + 1. The domain remains all real numbers because there are no operations causing any restrictions. The consistent domain in this case simplifies the understanding, but it sets the stage for the more complex domains we will encounter as we look at multiplication and division. The concept of the domain must be properly grasped, because any incorrect interpretation will lead to incorrect answers.

Finding fg (The Product of Functions)

Time for multiplication! Finding fg means we are going to multiply f(x) by g(x). We multiply the expressions of the two functions. So, we multiply (5x - 6) * (x - 7). To do this, we use the FOIL method (First, Outer, Inner, Last), or the distributive property.

• First: 5x * x = 5x².
• Outer: 5x * -7 = -35x.
• Inner: -6 * x = -6x.
• Last: -6 * -7 = 42.

Now, we combine all of these terms to obtain 5x² - 35x - 6x + 42. Combining the like terms -35x and -6x, we get -41x. Therefore, fg = 5x² - 41x + 42. Notice this is a quadratic function (because of the x² term), which makes it a bit more interesting!

Now, let's figure out the domain. For the product fg = 5x² - 41x + 42, there are still no restrictions on the x values. It is a polynomial function, and polynomial functions are defined for all real numbers. We can plug in any real number and get a real number back. Thus, the domain of fg is also all real numbers, or (-∞, ∞). It is crucial to remember that the domain of the product depends on the functions themselves. Understanding how domain interacts with function operations is essential as we progress.

To summarize: To find fg, we multiply the expressions of f(x) and g(x). With our functions, f(x) = 5x - 6 and g(x) = x - 7, the product is 5x² - 41x + 42. The domain remains all real numbers. Remember, because both functions are polynomials, their product is also a polynomial, and thus there are no restrictions. With the introduction of different function operations, it's increasingly essential to understand and be able to correctly identify the domain for each case. The domain plays a critical role in the definition and interpretation of mathematical functions, ensuring that results stay valid.

Finding f/g (The Quotient of Functions)

Finally, let's explore the quotient, f/g. This means we are going to divide f(x) by g(x). So, we have f(x) / g(x) = (5x - 6) / (x - 7). This is where things get a little different, guys! When we have a fraction, we need to be extra careful about the denominator. We cannot divide by zero. So, we need to find out what x value(s) would make the denominator, g(x) = x - 7, equal to zero. To do this, we set the denominator equal to zero and solve for x: x - 7 = 0. Adding 7 to both sides, we get x = 7. This means that x cannot be 7, because if x = 7, then the denominator becomes zero, and we'd be dividing by zero, which is not allowed.

Now, to determine the domain of f/g, we need to exclude this value. The domain of f/g is all real numbers except for x = 7. We can write this in a couple of ways:

• In interval notation: (-∞, 7) ∪ (7, ∞). This means all real numbers from negative infinity to 7 (but not including 7), and all real numbers from 7 to positive infinity (but not including 7).
• In set-builder notation: {x | x ≠ 7}. This means the set of all x such that x is not equal to 7.

So, for the quotient, the domain is restricted because of the division by zero. We've got to exclude the value that makes the denominator zero. This is a very important point!

To recap: to find f/g, we divide the expression of f(x) by the expression of g(x). For our functions, f(x) = 5x - 6 and g(x) = x - 7, the quotient is (5x - 6) / (x - 7). The domain is all real numbers except for x = 7, which we write as (-∞, 7) ∪ (7, ∞). The concept of domain becomes very apparent when performing division. It is essential to ensure that the denominator is never equal to zero. If you don't keep an eye on this you could get yourself in trouble! This is the most crucial takeaway from this whole article.

Summary

Alright, let's review what we've covered today! We've learned how to perform basic operations on functions: addition, subtraction, multiplication, and division. We've seen that:

• For f + g: add the expressions of f(x) and g(x). Domain: (-∞, ∞).
• For f - g: subtract the expression of g(x) from f(x). Domain: (-∞, ∞).
• For fg: multiply the expressions of f(x) and g(x). Domain: (-∞, ∞).
• For f/g: divide the expression of f(x) by the expression of g(x). Domain: (-∞, 7) ∪ (7, ∞).

And most importantly, we've learned how to find the domain of the resulting functions, understanding how different operations can affect the permissible x values. Remember, the domain is all the x values that are allowed to be plugged into a function. It's super important, especially when dealing with division! Keep practicing, and you'll become a function guru in no time. Thanks for hanging out, and keep math-ing!