Use The Given Functions To Find, Simplify, And Identify The Domain Of The Following Combinations.${ F(x) = \frac{5x + 1}{x + 7} \quad \text{and} \quad G(x) = \frac{-x - 6}{x + 8} }$1. { (f+g)(x) =$}$ { \square$}$

by ADMIN 214 views

In mathematics, functions are used to represent relationships between variables. When we combine two functions, we can create a new function that represents a more complex relationship. In this article, we will explore how to combine two functions, simplify the resulting expression, and identify the domain of the new function.

Understanding the Given Functions

We are given two functions:

  • f(x) = (5x + 1) / (x + 7): This function represents a linear relationship between x and y, where y is the result of dividing 5x + 1 by x + 7.
  • g(x) = (-x - 6) / (x + 8): This function represents a linear relationship between x and y, where y is the result of dividing -x - 6 by x + 8.

Combining Functions

To combine these two functions, we will add them together to create a new function, (f+g)(x).

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

Using the given functions, we can write the combined function as:

(f+g)(x) = (5x + 1) / (x + 7) + (-x - 6) / (x + 8)

To simplify this expression, we need to find a common denominator. The least common multiple of (x + 7) and (x + 8) is (x + 7)(x + 8).

(f+g)(x) = ((5x + 1)(x + 8) + (-x - 6)(x + 7)) / ((x + 7)(x + 8))

Expanding the numerator, we get:

(f+g)(x) = (5x^2 + 40x + 8 + -x^2 - 7x - 42) / ((x + 7)(x + 8))

Simplifying the numerator, we get:

(f+g)(x) = (4x^2 + 33x - 34) / ((x + 7)(x + 8))

Simplifying the Expression

To simplify the expression further, we can factor the numerator.

(f+g)(x) = ((4x - 17)(x + 2)) / ((x + 7)(x + 8))

Identifying the Domain

The domain of a function is the set of all possible input values (x) for which the function is defined. In this case, the function is defined for all real numbers except when the denominator is equal to zero.

(x + 7)(x + 8) ≠ 0

Solving for x, we get:

x ≠ -7 and x ≠ -8

Therefore, the domain of the combined function (f+g)(x) is all real numbers except -7 and -8.

Conclusion

In this article, we combined two functions, simplified the resulting expression, and identified the domain of the new function. We learned how to add functions together, find a common denominator, and simplify the resulting expression. We also learned how to identify the domain of a function by finding the values of x that make the denominator equal to zero.

Key Takeaways

  • To combine functions, add them together and find a common denominator.
  • To simplify an expression, expand the numerator and simplify the resulting expression.
  • To identify the domain of a function, find the values of x that make the denominator equal to zero.

Practice Problems

  1. Combine the functions f(x) = (2x + 3) / (x + 2) and g(x) = (-x + 1) / (x + 3) to create a new function.
  2. Simplify the resulting expression and identify the domain of the new function.
  3. Combine the functions f(x) = (x^2 + 2x + 1) / (x + 1) and g(x) = (-x^2 + 3x - 2) / (x - 1) to create a new function.
  4. Simplify the resulting expression and identify the domain of the new function.

References

In our previous article, we explored how to combine two functions, simplify the resulting expression, and identify the domain of the new function. In this article, we will answer some frequently asked questions about combining functions and simplifying expressions.

Q: What is the difference between combining functions and adding functions?

A: Combining functions involves adding or subtracting two or more functions to create a new function. Adding functions, on the other hand, involves combining two or more functions using the addition operation (+). For example, if we have two functions f(x) and g(x), combining them would result in a new function (f+g)(x), while adding them would result in a new function f(x) + g(x).

Q: How do I simplify a combined function?

A: To simplify a combined function, you need to find a common denominator and then combine the numerators. You can also use algebraic techniques such as factoring and canceling to simplify the expression.

Q: What is the domain of a combined function?

A: The domain of a combined function is the set of all possible input values (x) for which the function is defined. In general, the domain of a combined function is the intersection of the domains of the individual functions.

Q: How do I identify the domain of a combined function?

A: To identify the domain of a combined function, you need to find the values of x that make the denominator equal to zero. You can do this by setting the denominator equal to zero and solving for x.

Q: Can I combine more than two functions?

A: Yes, you can combine more than two functions. However, the process becomes more complex as the number of functions increases. You need to find a common denominator and then combine the numerators.

Q: What are some common mistakes to avoid when combining functions?

A: Some common mistakes to avoid when combining functions include:

  • Not finding a common denominator
  • Not combining the numerators correctly
  • Not simplifying the expression
  • Not identifying the domain of the combined function

Q: How do I use technology to simplify and identify the domain of a combined function?

A: You can use technology such as graphing calculators or computer algebra systems to simplify and identify the domain of a combined function. These tools can help you find a common denominator, combine the numerators, and identify the domain of the combined function.

Q: Can I use combining functions to solve real-world problems?

A: Yes, combining functions can be used to solve real-world problems. For example, you can use combining functions to model population growth, economic systems, or physical systems.

Q: What are some applications of combining functions in real-world problems?

A: Some applications of combining functions in real-world problems include:

  • Modeling population growth
  • Economic modeling
  • Physical modeling
  • Engineering design

Conclusion

In this article, we answered some frequently asked questions about combining functions and simplifying expressions. We discussed the difference between combining functions and adding functions, how to simplify a combined function, and how to identify the domain of a combined function. We also discussed some common mistakes to avoid when combining functions and how to use technology to simplify and identify the domain of a combined function.

Key Takeaways

  • Combining functions involves adding or subtracting two or more functions to create a new function.
  • Simplifying a combined function involves finding a common denominator and combining the numerators.
  • Identifying the domain of a combined function involves finding the values of x that make the denominator equal to zero.
  • Technology can be used to simplify and identify the domain of a combined function.

Practice Problems

  1. Combine the functions f(x) = (2x + 3) / (x + 2) and g(x) = (-x + 1) / (x + 3) to create a new function.
  2. Simplify the resulting expression and identify the domain of the new function.
  3. Combine the functions f(x) = (x^2 + 2x + 1) / (x + 1) and g(x) = (-x^2 + 3x - 2) / (x - 1) to create a new function.
  4. Simplify the resulting expression and identify the domain of the new function.

References