Composite Function: Find (g O F)(-3) Value

Understanding composite functions can seem tricky at first, but once you grasp the core concept, you'll be solving these problems with ease. This article will walk you through a step-by-step solution to finding the value of a composite function, specifically $(g \circ f)(-3)$, given the functions $f(x) = -7x + 6$ and $g(x) = -6x^2 - 6x + 5$. So, let's dive in and unravel this mathematical puzzle together!

What are Composite Functions?

Before we jump into the problem, let's briefly discuss composite functions. A composite function is essentially a function within a function. We denote it as $(g \circ f)(x)$, which is read as "g of f of x." This means we first apply the function $f$ to $x$, and then we take the result and plug it into the function $g$. Think of it like a two-step process: input $x$, process it through $f$, take that output, and process it through $g$. This sequential operation is the heart of composite functions.

Understanding this concept is crucial for solving problems like the one we have. The notation $(g \circ f)(-3)$ tells us we need to first find the value of $f(-3)$, and then use that value as the input for the function $g$. By breaking down the problem into smaller steps, we can approach it methodically and avoid confusion. The order of operations is key here, so always start with the innermost function and work your way outwards. With a clear grasp of this principle, composite function problems become much less daunting and more manageable.

Step 1: Evaluate f(-3)

Our first task is to find the value of $f(-3)$. We're given the function $f(x) = -7x + 6$. To evaluate $f(-3)$, we simply substitute $-3$ for $x$ in the expression. This means replacing every instance of $x$ with the value $-3$. Careful substitution is crucial to avoid errors, so double-check that you've correctly placed the value in the expression.

So, we have:

$f(-3) = -7(-3) + 6$

Now, we perform the arithmetic. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this case, we first perform the multiplication:

$f(-3) = 21 + 6$

Then, we perform the addition:

$f(-3) = 27$

Therefore, the value of $f(-3)$ is 27. This is a crucial intermediate result, as it becomes the input for our next step. Make sure you've correctly calculated this value before moving on, as an error here will propagate through the rest of the solution. With $f(-3)$ in hand, we're ready to tackle the second part of the problem.

Step 2: Evaluate g(f(-3))

Now that we know $f(-3) = 27$, we can move on to the second part of the problem: finding $g(f(-3))$. Remember, $(g \circ f)(-3)$ means we need to evaluate $g$ at the value we just found for $f(-3)$. So, we're essentially looking for $g(27)$. This is where the composite function concept truly shines, as we're using the output of one function as the input for another.

We're given the function $g(x) = -6x^2 - 6x + 5$. To find $g(27)$, we substitute 27 for $x$ in this expression, just like we did with $f(x)$. Again, meticulous substitution is key to avoid mistakes. Double-check that you've replaced all instances of $x$ with 27.

So, we have:

$g(27) = -6(27)^2 - 6(27) + 5$

Now comes the arithmetic, and it's a bit more involved this time. We need to follow the order of operations carefully. First, we evaluate the exponent:

$g(27) = -6(729) - 6(27) + 5$

Next, we perform the multiplications:

$g(27) = -4374 - 162 + 5$

Finally, we perform the additions and subtractions from left to right:

$g(27) = -4536 + 5$

$g(27) = -4531$

Therefore, the value of $g(f(-3))$, or $(g \circ f)(-3)$, is -4531. This is our final answer. The calculations here are more complex, so it's always a good idea to double-check your work, especially the multiplication and subtraction steps. With this final result, we've successfully navigated the composite function and found its value at the specified point.

Step 3: Final Answer

After carefully evaluating $f(-3)$ and then using that result to evaluate $g(f(-3))$, we've arrived at our final answer. We found that $(g \circ f)(-3) = -4531$. This means that when we first apply the function $f$ to -3, and then apply the function $g$ to the result, we get -4531.

This process highlights the essence of composite functions: a chain reaction where the output of one function becomes the input of another. Understanding this sequential operation is crucial for mastering composite functions. By breaking down the problem into smaller, manageable steps, as we did here, you can tackle even complex composite function evaluations with confidence.

Remember to always double-check your work, especially the arithmetic, to ensure accuracy. Composite functions are a fundamental concept in mathematics, and mastering them will be invaluable as you progress in your studies. With practice and a clear understanding of the process, you'll be able to solve these problems efficiently and accurately.

Conclusion

In this article, we've walked through a detailed solution for finding the value of the composite function $(g \circ f)(-3)$, given $f(x) = -7x + 6$ and $g(x) = -6x^2 - 6x + 5$. We broke the problem down into manageable steps: first, evaluating $f(-3)$, and then using that result as the input for $g(x)$. This step-by-step approach is key to understanding and solving composite function problems.

We found that $f(-3) = 27$, and subsequently, $g(27) = -4531$. Therefore, $(g \circ f)(-3) = -4531$. This final answer represents the value of the composite function at the specific input of -3.

Understanding composite functions is a crucial step in your mathematical journey. They appear in various areas of mathematics and are essential for building a strong foundation. By practicing and applying these concepts, you'll become more proficient in solving a wide range of mathematical problems.

Remember to always double-check your work and break down complex problems into smaller, more manageable steps. With a clear understanding of the underlying principles and a methodical approach, you can confidently tackle any composite function challenge.

For further exploration of composite functions and related topics, you might find the resources at Khan Academy helpful.