Evaluating Function Combinations

Oct 13, 2025 by miodeya.com 135 views

Understanding Function Combinations

When dealing with function combinations, it's essential to understand how different functions interact with each other. In this case, we have two functions: $f(x)=x^2-6x-27$ and $g(x)=x-9$ . Our goal is to match the given expressions with the correct function combinations.

Function Combinations

A function combination is the result of applying one or more functions to an input value. In this case, we can create new functions by combining $f(x)$ and $g(x)$ in various ways. The possible function combinations are:

$f(g(x))$
$g(f(x))$
$(f \circ g)(x)$
$(g \circ f)(x)$

Evaluating Function Combinations

To evaluate a function combination, we need to apply the inner function first and then apply the outer function to the result. Let's consider the first combination, $f(g(x))$ . We start by evaluating $g(x)$ , which gives us $x-9$ . Then, we substitute this result into $f(x)$ , giving us $f(x-9)$ . Expanding this expression, we get:

$f(g(x)) = (x-9)^2 - 6(x-9) - 27$

Expanding and simplifying this expression, we get:

$f(g(x)) = x^2 - 18x + 81 - 6x + 54 - 27$

Combining like terms, we get:

$f(g(x)) = x^2 - 24x + 108$

Matching Expressions to Function Combinations

Now that we have evaluated the function combinations, we can match the given expressions with the correct function combinations. The expressions are:

$x^2 - 24x + 108$
$x^2 - 18x + 81$
$x^2 - 6x - 27$
$x - 9$

Matching Expressions

Expression 1: $x^2 - 24x + 108$ This expression matches with the function combination $f(g(x))$ . We can verify this by comparing the coefficients and constant terms.
Expression 2: $x^2 - 18x + 81$ This expression matches with the function combination $g(f(x))$ . We can verify this by comparing the coefficients and constant terms.
Expression 3: $x^2 - 6x - 27$ This expression matches with the function $f(x)$ . We can verify this by comparing the coefficients and constant terms.
Expression 4: $x - 9$ This expression matches with the function $g(x)$ . We can verify this by comparing the coefficients and constant terms.

Conclusion

In this article, we have matched the given expressions with the correct function combinations. We have evaluated the function combinations and compared the coefficients and constant terms to determine the correct matches. This exercise has helped us understand how different functions interact with each other and how to evaluate function combinations.

Discussion

What are some common function combinations that we encounter in real-world applications?
How do we evaluate function combinations in different mathematical contexts?
Can you think of any other function combinations that we can create using $f(x)$ and $g(x)$ ?

Practice Problems

Evaluate the function combination $f(g(x))$ for the given function $f(x) = x^2 - 6x - 27$ and $g(x) = x - 9$ .
Match the expressions with the correct function combinations:
- $x^2 - 18x + 81$
- $x^2 - 6x - 27$
- $x - 9$
- $x^2 - 24x + 108$

Answer Key

$f(g(x)) = x^2 - 24x + 108$
Expression 1: $x^2 - 24x + 108$ matches with $f(g(x))$
Expression 2: $x^2 - 18x + 81$ matches with $g(f(x))$
Expression 3: $x^2 - 6x - 27$ matches with $f(x)$
Expression 4: $x - 9$ matches with $g(x)$

Q&A: Function Combinations

In this article, we will answer some frequently asked questions about function combinations. If you have any questions or need further clarification, feel free to ask.

Q: What is a function combination?

A: A function combination is the result of applying one or more functions to an input value. In this case, we can create new functions by combining $f(x)$ and $g(x)$ in various ways.

Q: How do we evaluate a function combination?

A: To evaluate a function combination, we need to apply the inner function first and then apply the outer function to the result. For example, to evaluate $f(g(x))$ , we start by evaluating $g(x)$ , which gives us $x-9$ . Then, we substitute this result into $f(x)$ , giving us $f(x-9)$ .

Q: What are some common function combinations that we encounter in real-world applications?

A: Some common function combinations that we encounter in real-world applications include:

Composition of functions: This is a function combination where one function is applied to the output of another function.
Inverse functions: This is a function combination where one function is the inverse of another function.
Function multiplication: This is a function combination where one function is multiplied by another function.

Q: How do we match expressions with function combinations?

A: To match expressions with function combinations, we need to compare the coefficients and constant terms of the expressions with the function combinations. For example, if we have the expression $x^2 - 24x + 108$ , we can match it with the function combination $f(g(x))$ by comparing the coefficients and constant terms.

Q: Can you think of any other function combinations that we can create using $f(x)$ and $g(x)$ ?

A: Yes, there are many other function combinations that we can create using $f(x)$ and $g(x)$ . Some examples include:

$(f \circ g)(x)$ : This is a function combination where $f(x)$ is applied to the output of $g(x)$ .
$(g \circ f)(x)$ : This is a function combination where $g(x)$ is applied to the output of $f(x)$ .
$f(f(x))$ : This is a function combination where $f(x)$ is applied to the output of $f(x)$ .

Q: How do we evaluate function combinations in different mathematical contexts?

A: To evaluate function combinations in different mathematical contexts, we need to consider the specific rules and properties of the functions involved. For example, if we are working with trigonometric functions, we need to consider the properties of sine, cosine, and tangent.

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

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

Not following the order of operations: When evaluating a function combination, we need to follow the order of operations (PEMDAS) to ensure that we are applying the functions in the correct order.
Not considering the domain and range: When working with function combinations, we need to consider the domain and range of the functions involved to ensure that we are working with valid input values.
Not simplifying expressions: When working with function combinations, we need to simplify expressions to ensure that we are working with the simplest possible form.

Conclusion

In this article, we have answered some frequently asked questions about function combinations. We have discussed the basics of function combinations, how to evaluate them, and how to match expressions with function combinations. We have also discussed some common function combinations that we encounter in real-world applications and some common mistakes to avoid when working with function combinations.

Discussion

What are some other function combinations that we can create using $f(x)$ and $g(x)$ ?
How do we evaluate function combinations in different mathematical contexts?
What are some common mistakes to avoid when working with function combinations?

Practice Problems

Evaluate the function combination $f(g(x))$ for the given function $f(x) = x^2 - 6x - 27$ and $g(x) = x - 9$ .
Match the expressions with the correct function combinations:
- $x^2 - 18x + 81$
- $x^2 - 6x - 27$
- $x - 9$
- $x^2 - 24x + 108$

Answer Key

$f(g(x)) = x^2 - 24x + 108$
Expression 1: $x^2 - 24x + 108$ matches with $f(g(x))$
Expression 2: $x^2 - 18x + 81$ matches with $g(f(x))$
Expression 3: $x^2 - 6x - 27$ matches with $f(x)$
Expression 4: $x - 9$ matches with $g(x)$