Let $f(x) = 4x + 5$ And $g(x) = 2x - 7$. Find $(f+g)(x$\], $(f-g)(x$\], $(fg)(x$\], And $\left(\frac{f}{g}\right)(x$\]. Give The Domain Of Each.1. $(f+g)(x) = 6x - 2$ Simplify Your Answer.

Let's Dive into the World of Functions: Combining and Simplifying

In the realm of mathematics, functions play a vital role in representing relationships between variables. When dealing with multiple functions, it's often necessary to combine them to create new functions. In this article, we'll explore the process of combining two given functions, $f(x) = 4x + 5$ and $g(x) = 2x - 7$, to find the sum, difference, product, and quotient of these functions. We'll also determine the domain of each resulting function.

To combine functions, we'll use the following rules:

• Sum of Functions: $(f+g)(x) = f(x) + g(x)$
• Difference of Functions: $(f-g)(x) = f(x) - g(x)$
• Product of Functions: $(fg)(x) = f(x) \cdot g(x)$
• Quotient of Functions: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$

Finding the Sum of Functions

To find the sum of functions $(f+g)(x)$, we'll add the corresponding terms of $f(x)$ and $g(x)$.

Step 1: Add the corresponding terms of $f(x)$ and $g(x)$

$(f+g)(x) = (4x + 5) + (2x - 7)$

Step 2: Simplify the expression

$(f+g)(x) = 4x + 5 + 2x - 7$

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

Finding the Difference of Functions

To find the difference of functions $(f-g)(x)$, we'll subtract the corresponding terms of $g(x)$ from $f(x)$.

Step 1: Subtract the corresponding terms of $g(x)$ from $f(x)$

$(f-g)(x) = (4x + 5) - (2x - 7)$

Step 2: Simplify the expression

$(f-g)(x) = 4x + 5 - 2x + 7$

$(f-g)(x) = 2x + 12$

Finding the Product of Functions

To find the product of functions $(fg)(x)$, we'll multiply the corresponding terms of $f(x)$ and $g(x)$.

Step 1: Multiply the corresponding terms of $f(x)$ and $g(x)$

$(fg)(x) = (4x + 5) \cdot (2x - 7)$

Step 2: Simplify the expression

$(fg)(x) = 8x^2 - 28x + 10x - 35$

$(fg)(x) = 8x^2 - 18x - 35$

Finding the Quotient of Functions

To find the quotient of functions $\left(\frac{f}{g}\right)(x)$, we'll divide the corresponding terms of $f(x)$ by $g(x)$.

Step 1: Divide the corresponding terms of $f(x)$ by $g(x)$

$\left(\frac{f}{g}\right)(x) = \frac{4x + 5}{2x - 7}$

Step 2: Simplify the expression

$\left(\frac{f}{g}\right)(x) = \frac{4x + 5}{2x - 7}$

To determine the domain of each function, we'll consider the restrictions imposed by the functions.

• Domain of $(f+g)(x)$: Since $(f+g)(x) = 6x - 2$, the domain is all real numbers, $(-\infty, \infty)$.
• Domain of $(f-g)(x)$: Since $(f-g)(x) = 2x + 12$, the domain is all real numbers, $(-\infty, \infty)$.
• Domain of $(fg)(x)$: Since $(fg)(x) = 8x^2 - 18x - 35$, the domain is all real numbers, $(-\infty, \infty)$.
• Domain of $\left(\frac{f}{g}\right)(x)$: Since $\left(\frac{f}{g}\right)(x) = \frac{4x + 5}{2x - 7}$, the domain is all real numbers except where $2x - 7 = 0$, which is $x = \frac{7}{2}$. Therefore, the domain is $(-\infty, \frac{7}{2}) \cup (\frac{7}{2}, \infty)$.

In conclusion, we've successfully combined the functions $f(x) = 4x + 5$ and $g(x) = 2x - 7$ to find the sum, difference, product, and quotient of these functions. We've also determined the domain of each resulting function.
Let's Dive into the World of Functions: Combining and Simplifying

Q: What is the sum of functions $(f+g)(x)$?

A: The sum of functions $(f+g)(x)$ is found by adding the corresponding terms of $f(x)$ and $g(x)$. In this case, $(f+g)(x) = (4x + 5) + (2x - 7) = 6x - 2$.

Q: What is the difference of functions $(f-g)(x)$?

A: The difference of functions $(f-g)(x)$ is found by subtracting the corresponding terms of $g(x)$ from $f(x)$. In this case, $(f-g)(x) = (4x + 5) - (2x - 7) = 2x + 12$.

Q: What is the product of functions $(fg)(x)$?

A: The product of functions $(fg)(x)$ is found by multiplying the corresponding terms of $f(x)$ and $g(x)$. In this case, $(fg)(x) = (4x + 5) \cdot (2x - 7) = 8x^2 - 18x - 35$.

Q: What is the quotient of functions $\left(\frac{f}{g}\right)(x)$?

A: The quotient of functions $\left(\frac{f}{g}\right)(x)$ is found by dividing the corresponding terms of $f(x)$ by $g(x)$. In this case, $\left(\frac{f}{g}\right)(x) = \frac{4x + 5}{2x - 7}$.

Q: What is the domain of each function?

A: The domain of each function is as follows:

• Domain of $(f+g)(x)$: All real numbers, $(-\infty, \infty)$.
• Domain of $(f-g)(x)$: All real numbers, $(-\infty, \infty)$.
• Domain of $(fg)(x)$: All real numbers, $(-\infty, \infty)$.
• Domain of $\left(\frac{f}{g}\right)(x)$: All real numbers except where $2x - 7 = 0$, which is $x = \frac{7}{2}$. Therefore, the domain is $(-\infty, \frac{7}{2}) \cup (\frac{7}{2}, \infty)$.

Q: Why is it important to determine the domain of each function?

A: Determining the domain of each function is crucial because it helps us understand the restrictions imposed by the functions. This, in turn, affects the validity of the function's output.

Q: Can you provide examples of real-world applications of combining functions?

A: Yes, combining functions has numerous real-world applications. For instance:

• Physics: When modeling the motion of an object, we often combine functions to represent the object's position, velocity, and acceleration.
• Economics: In economics, we use functions to model the behavior of markets, consumers, and producers. Combining functions helps us understand the relationships between these entities.
• Computer Science: In computer science, we use functions to represent algorithms and data structures. Combining functions helps us create more complex and efficient algorithms.

Q: How can I practice combining functions?

A: To practice combining functions, try the following:

• Work through examples: Start with simple examples and gradually move on to more complex ones.
• Use online resources: Websites like Khan Academy, Mathway, and Wolfram Alpha offer interactive tools and exercises to help you practice combining functions.
• Join a study group: Joining a study group or finding a study buddy can help you stay motivated and learn from others.

By following these tips and practicing regularly, you'll become proficient in combining functions and be able to tackle more complex problems with confidence.