Given Expressions and Function Combinations

Oct 13, 2025 by miodeya.com 78 views

Understanding Function Combinations and Expressions

In mathematics, function combinations and expressions are essential concepts that help us understand how functions interact with each other. When we have two functions, we can combine them in various ways to create new functions. In this article, we will explore the concept of function combinations and expressions, and we will match the given expressions to the correct function combinations.

Function Combinations

Function combinations refer to the process of combining two or more functions to create a new function. This can be done in various ways, such as addition, subtraction, multiplication, and division. When we combine functions, we need to follow the order of operations (PEMDAS) to ensure that the correct order of operations is maintained.

Function Expressions

Function expressions, on the other hand, refer to the representation of a function using mathematical notation. A function expression is a mathematical statement that describes the relationship between the input and output of a function. Function expressions can be represented using various notations, such as algebraic expressions, trigonometric expressions, and exponential expressions.

Given Expressions and Function Combinations

We are given two functions:

\begin{array}{l} f(x)=x^2-6x-27 \\ g(x)=x-9 \end{array}

We need to match the given expressions to the correct function combinations. The given expressions are:

$f(x) + g(x)$
$f(x) - g(x)$
$f(x) \cdot g(x)$
$f(x) / g(x)$

Matching Expressions to Function Combinations

To match the expressions to the correct function combinations, we need to evaluate each expression using the given functions.

Expression 1: $f(x) + g(x)$

To evaluate this expression, we need to add the two functions:

\begin{align*} f(x) + g(x) &= (x^2-6x-27) + (x-9) \\ &= x^2-6x-27+x-9 \\ &= x^2-5x-36 \end{align*}

Expression 2: $f(x) - g(x)$

To evaluate this expression, we need to subtract the two functions:

\begin{align*} f(x) - g(x) &= (x^2-6x-27) - (x-9) \\ &= x^2-6x-27-x+9 \\ &= x^2-7x-18 \end{align*}

Expression 3: $f(x) \cdot g(x)$

To evaluate this expression, we need to multiply the two functions:

\begin{align*} f(x) \cdot g(x) &= (x^2-6x-27) \cdot (x-9) \\ &= x^2(x-9)-6x(x-9)-27(x-9) \\ &= x^3-9x^2-6x^2+54x-27x+243 \\ &= x^3-15x^2+27x+243 \end{align*}

Expression 4: $f(x) / g(x)$

To evaluate this expression, we need to divide the two functions:

\begin{align*} f(x) / g(x) &= \frac{x^2-6x-27}{x-9} \\ &= \frac{(x-9)(x+3)}{x-9} \\ &= x+3 \end{align*}

Conclusion

In this article, we explored the concept of function combinations and expressions, and we matched the given expressions to the correct function combinations. We evaluated each expression using the given functions and obtained the following results:

$f(x) + g(x) = x^2-5x-36$
$f(x) - g(x) = x^2-7x-18$
$f(x) \cdot g(x) = x^3-15x^2+27x+243$
$f(x) / g(x) = x+3$

We hope that this article has provided a clear understanding of function combinations and expressions, and we hope that it has helped you to match the expressions to the correct function combinations.

Understanding Function Combinations and Expressions: Q&A

In our previous article, we explored the concept of function combinations and expressions, and we matched the given expressions to the correct function combinations. In this article, we will answer some frequently asked questions about function combinations and expressions.

Q1: What is the difference between function combinations and function expressions?

A1: Function combinations refer to the process of combining two or more functions to create a new function. Function expressions, on the other hand, refer to the representation of a function using mathematical notation.

Q2: How do I evaluate a function combination?

A2: To evaluate a function combination, you need to follow the order of operations (PEMDAS) and combine the functions using the correct mathematical operations (addition, subtraction, multiplication, and division).

Q3: What is the purpose of function combinations?

A3: Function combinations are used to create new functions by combining existing functions. This can be useful in a variety of mathematical applications, such as solving equations, graphing functions, and modeling real-world phenomena.

Q4: How do I match expressions to function combinations?

A4: To match expressions to function combinations, you need to evaluate each expression using the given functions and compare the results to the correct function combinations.

Q5: What are some common function combinations?

A5: Some common function combinations include:

Addition: $f(x) + g(x)$
Subtraction: $f(x) - g(x)$
Multiplication: $f(x) \cdot g(x)$
Division: $f(x) / g(x)$

Q6: How do I simplify a function combination?

A6: To simplify a function combination, you need to combine like terms and eliminate any unnecessary operations.

Q7: What are some real-world applications of function combinations?

A7: Function combinations have a wide range of real-world applications, including:

Modeling population growth and decline
Analyzing financial data and predicting stock prices
Solving optimization problems in engineering and economics
Graphing functions and visualizing data

Q8: How do I use function combinations in calculus?

A8: Function combinations are used extensively in calculus to solve optimization problems, find derivatives, and integrate functions.

Q9: What are some common mistakes to avoid when working with function combinations?

A9: Some common mistakes to avoid when working with function combinations include:

Failing to follow the order of operations (PEMDAS)
Not combining like terms
Not eliminating unnecessary operations
Not checking for domain restrictions

Q10: How do I practice function combinations?

A10: To practice function combinations, you can try the following:

Work through example problems and exercises
Use online resources and calculators to visualize and explore function combinations
Practice simplifying and evaluating function combinations
Apply function combinations to real-world problems and scenarios

Conclusion

In this article, we answered some frequently asked questions about function combinations and expressions. We hope that this article has provided a clear understanding of function combinations and expressions, and we hope that it has helped you to practice and apply function combinations in a variety of mathematical and real-world contexts.