Let $f(x) = X^2 - 6$ And $g(x) = 18 - X$. Perform The Composition Or Operation Indicated.Find $(f \cdot G)(-9$\].$(f \cdot G)(-9) = \, \square$

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Introduction

In mathematics, composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. Given two functions, $f(x)$ and $g(x)$, the composition of $f$ and $g$ is denoted by $(f \cdot g)(x)$ and is defined as $(f \cdot g)(x) = f(g(x))$. In this article, we will explore the composition of two given functions, $f(x) = x^2 - 6$ and $g(x) = 18 - x$, and find the value of $(f \cdot g)(-9)$.

Composition of Functions

To find the composition of $f$ and $g$, we need to substitute $g(x)$ into $f(x)$ in place of $x$. This means that we will replace every instance of $x$ in the function $f(x)$ with the function $g(x)$. The resulting function is then denoted by $(f \cdot g)(x)$.

Finding $(f \cdot g)(x)$

To find the composition of $f$ and $g$, we will substitute $g(x) = 18 - x$ into $f(x) = x^2 - 6$ in place of $x$. This gives us:

$(f \cdot g)(x) = f(g(x)) = (18 - x)^2 - 6$

Expanding the Expression

To simplify the expression, we will expand the squared term using the formula $(a - b)^2 = a^2 - 2ab + b^2$. In this case, $a = 18$ and $b = x$. This gives us:

$(f \cdot g)(x) = (18)^2 - 2(18)(x) + (x)^2 - 6$

Simplifying the Expression

Now, we will simplify the expression by combining like terms. This gives us:

$(f \cdot g)(x) = 324 - 36x + x^2 - 6$

Combining Like Terms

Next, we will combine the constant terms by adding $324$ and $-6$. This gives us:

$(f \cdot g)(x) = 318 - 36x + x^2$

Finding $(f \cdot g)(-9)$

Now that we have found the composition of $f$ and $g$, we can find the value of $(f \cdot g)(-9)$ by substituting $x = -9$ into the expression. This gives us:

$(f \cdot g)(-9) = 318 - 36(-9) + (-9)^2$

Evaluating the Expression

To evaluate the expression, we will first simplify the squared term. This gives us:

$(-9)^2 = 81$

Substituting the Value

Now, we will substitute the value of $(-9)^2$ into the expression. This gives us:

$(f \cdot g)(-9) = 318 - 36(-9) + 81$

Evaluating the Expression

Next, we will simplify the expression by multiplying $36$ and $-9$. This gives us:

$36(-9) = -324$

Substituting the Value

Now, we will substitute the value of $36(-9)$ into the expression. This gives us:

$(f \cdot g)(-9) = 318 + 324 + 81$

Evaluating the Expression

Finally, we will simplify the expression by adding the constant terms. This gives us:

$(f \cdot g)(-9) = 723$

The final answer is $\boxed{723}$.

Introduction

In our previous article, we explored the composition of two functions, $f(x) = x^2 - 6$ and $g(x) = 18 - x$, and found the value of $(f \cdot g)(-9)$. In this article, we will answer some frequently asked questions about composition of functions.

Q1: What is the composition of functions?

A1: The composition of functions is a way of combining two or more functions to create a new function. Given two functions, $f(x)$ and $g(x)$, the composition of $f$ and $g$ is denoted by $(f \cdot g)(x)$ and is defined as $(f \cdot g)(x) = f(g(x))$.

Q2: How do I find the composition of two functions?

A2: To find the composition of two functions, you need to substitute one function into the other in place of the variable. For example, if we want to find the composition of $f(x) = x^2 - 6$ and $g(x) = 18 - x$, we will substitute $g(x)$ into $f(x)$ in place of $x$. This gives us $(f \cdot g)(x) = f(g(x)) = (18 - x)^2 - 6$.

Q3: What is the difference between function composition and function evaluation?

A3: Function composition is a way of combining two or more functions to create a new function, while function evaluation is the process of finding the value of a function at a given input. For example, if we want to find the value of $(f \cdot g)(-9)$, we need to evaluate the function $(f \cdot g)(x) = (18 - x)^2 - 6$ at $x = -9$.

Q4: Can I compose more than two functions?

A4: Yes, you can compose more than two functions. For example, if we have three functions, $f(x)$, $g(x)$, and $h(x)$, we can find the composition of $f$ and $g$ first, and then find the composition of the result with $h(x)$. This gives us $(f \cdot g \cdot h)(x) = f(g(h(x)))$.

Q5: What are some common applications of function composition?

A5: Function composition has many applications in mathematics, science, and engineering. Some common applications include:

• Modeling real-world phenomena, such as population growth or chemical reactions
• Solving systems of equations
• Finding the inverse of a function
• Creating new functions from existing ones

Q6: Can I use function composition to find the inverse of a function?

A6: Yes, you can use function composition to find the inverse of a function. If we have a function $f(x)$ and we want to find its inverse, we can find the composition of $f$ with its inverse, $f^{-1}(x)$. This gives us $(f \cdot f^{-1})(x) = f(f^{-1}(x)) = x$.

Q7: What are some common mistakes to avoid when working with function composition?

A7: Some common mistakes to avoid when working with function composition include:

• Not following the order of operations
• Not simplifying the expression before evaluating it
• Not checking for domain restrictions
• Not using the correct notation for function composition

Q8: Can I use function composition to solve systems of equations?

A8: Yes, you can use function composition to solve systems of equations. If we have two functions, $f(x)$ and $g(x)$, and we want to find the solution to the system of equations $f(x) = g(x)$, we can find the composition of $f$ and $g$ and set it equal to zero. This gives us $(f \cdot g)(x) = f(g(x)) = 0$.

Q9: What are some common tools and techniques used to work with function composition?

A9: Some common tools and techniques used to work with function composition include:

• Graphing calculators
• Computer algebra systems
• Mathematical software
• Algebraic manipulations

Q10: Can I use function composition to model real-world phenomena?

A10: Yes, you can use function composition to model real-world phenomena. For example, if we want to model the growth of a population, we can use a function that represents the population at a given time, and then use function composition to find the population at a later time.

The final answer is $\boxed{723}$.