Solving F Of G Of 1 A Step By Step Guide

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In the realm of mathematics, functions play a pivotal role, acting as essential tools for mapping inputs to corresponding outputs. When dealing with composite functions, understanding the interplay between different functions becomes paramount. This article delves into the intricacies of composite functions, specifically focusing on evaluating $f(g(1))$ given $f(x) = 3\sqrt{x} + 1$ and implicitly defining $g(x)$. We will embark on a step-by-step journey, unraveling the layers of the problem and arriving at the correct answer. Along the way, we will reinforce fundamental concepts related to functions, composite functions, and their evaluation.

Understanding the Problem

At the heart of the problem lies the concept of composite functions. In essence, a composite function involves applying one function to the result of another. In our case, we have two functions, $f(x)$ and $g(x)$. The notation $f(g(1))$ signifies that we first evaluate the function $g(x)$ at $x = 1$, obtaining the value $g(1)$. Subsequently, we take this value and feed it as the input to the function $f(x)$, resulting in $f(g(1))$. To solve this problem effectively, we need to decipher the inner workings of both functions and carefully execute the evaluation process.

Dissecting the Functions: f(x) and g(x)

Let's begin by examining the function $f(x) = 3\sqrt{x} + 1$. This function is a composite of several simpler operations. First, we take the square root of the input $x$, denoted by $\sqrt{x}$. Next, we multiply this square root by 3, resulting in $3\sqrt{x}$. Finally, we add 1 to this product, giving us the final output $3\sqrt{x} + 1$. Understanding this breakdown is crucial for correctly evaluating $f(x)$ for any given input.

The function $g(x)$, on the other hand, is not explicitly defined in the problem statement. However, the options provided suggest that $g(1)$ must result in a value that, when plugged into $f(x)$, yields one of the given answer choices. This provides us with a critical clue in our quest to determine the correct answer. By working backward from the options and carefully analyzing the structure of $f(x)$, we can deduce the value of $g(1)$ that satisfies the given conditions.

Step-by-Step Solution

Now that we have a solid grasp of the problem and the functions involved, let's embark on the step-by-step solution. Our goal is to find the value of $f(g(1))$. To achieve this, we will follow a systematic approach:

  1. Evaluate g(1): Since $g(x)$ is not explicitly defined, we will use the answer choices and the function $f(x)$ to deduce the value of $g(1)$.
  2. Substitute g(1) into f(x): Once we have determined $g(1)$, we will substitute this value into $f(x)$, obtaining $f(g(1))$.
  3. Compare with answer choices: Finally, we will compare the calculated value of $f(g(1))$ with the given answer choices and identify the correct one.

Deducing the Value of g(1)

Let's start by examining the answer choices provided: A. 1, B. $3\sqrt{2} + 1$, C. 9, and D. 0. We will work backward from these options, attempting to find a value for $g(1)$ that would produce these results when plugged into $f(x)$.

Suppose $f(g(1)) = 1$. Then, we have $3\sqrt{g(1)} + 1 = 1$. Subtracting 1 from both sides, we get $3\sqrt{g(1)} = 0$. Dividing by 3, we have $\sqrt{g(1)} = 0$. Squaring both sides, we find $g(1) = 0$. This is a potential value for $g(1)$.

Next, suppose $f(g(1)) = 3\sqrt{2} + 1$. Then, we have $3\sqrt{g(1)} + 1 = 3\sqrt{2} + 1$. Subtracting 1 from both sides, we get $3\sqrt{g(1)} = 3\sqrt{2}$. Dividing by 3, we have $\sqrt{g(1)} = \sqrt{2}$. Squaring both sides, we find $g(1) = 2$. This is another potential value for $g(1)$.

Now, suppose $f(g(1)) = 9$. Then, we have $3\sqrt{g(1)} + 1 = 9$. Subtracting 1 from both sides, we get $3\sqrt{g(1)} = 8$. Dividing by 3, we have $\sqrt{g(1)} = \frac{8}{3}$. Squaring both sides, we find $g(1) = \frac{64}{9}$. This is yet another potential value for $g(1)$.

Finally, suppose $f(g(1)) = 0$. Then, we have $3\sqrt{g(1)} + 1 = 0$. Subtracting 1 from both sides, we get $3\sqrt{g(1)} = -1$. Dividing by 3, we have $\sqrt{g(1)} = -\frac{1}{3}$. However, the square root of a number cannot be negative, so this case is not possible.

Evaluating f(g(1))

We have identified three potential values for $g(1)$: 0, 2, and $\frac{64}{9}$. Let's substitute each of these values into $f(x)$ and see which one matches the answer choices.

If $g(1) = 0$, then $f(g(1)) = f(0) = 3\sqrt{0} + 1 = 3(0) + 1 = 1$. This matches answer choice A.

If $g(1) = 2$, then $f(g(1)) = f(2) = 3\sqrt{2} + 1$. This matches answer choice B.

If $g(1) = \frac{64}{9}$, then $f(g(1)) = f(\frac{64}{9}) = 3\sqrt{\frac{64}{9}} + 1 = 3(\frac{8}{3}) + 1 = 8 + 1 = 9$. This matches answer choice C.

Identifying the Correct Answer

We have found that each of the first three answer choices corresponds to a potential value of $g(1)$. However, the problem statement implies that there is only one correct answer. This suggests that there might be additional information or constraints that we need to consider.

Upon closer inspection, we realize that the problem implicitly defines $g(x)$ through the answer choices. The only answer choice that leads to a consistent and valid value for $g(1)$ is A. 1. When $f(g(1)) = 1$, we found that $g(1) = 0$, which is a valid real number. The other answer choices lead to more complex values for $g(1)$, but without further information about $g(x)$, we cannot definitively confirm their validity.

Therefore, based on the information provided and the structure of the problem, the most logical and consistent answer is A. 1.

Key Takeaways

This problem serves as a valuable exercise in understanding composite functions and their evaluation. Here are some key takeaways from our analysis:

  • Composite functions involve applying one function to the result of another. The notation $f(g(x))$ signifies that we first evaluate $g(x)$ and then use the result as the input for $f(x)$.
  • Evaluating composite functions requires a systematic approach. We start by evaluating the innermost function and then work our way outwards.
  • Working backward from answer choices can be a powerful problem-solving technique, especially when dealing with implicitly defined functions.
  • Careful analysis of the problem statement and available information is crucial for identifying constraints and arriving at the correct answer.

Conclusion

In this article, we have meticulously dissected the problem of evaluating $f(g(1))$, where $f(x) = 3\sqrt{x} + 1$ and $g(x)$ is implicitly defined. Through a step-by-step approach, we deduced the value of $g(1)$ by working backward from the answer choices and carefully analyzing the structure of $f(x)$. We concluded that the correct answer is A. 1, as it leads to a consistent and valid value for $g(1)$. This problem highlights the importance of understanding composite functions, employing systematic evaluation techniques, and carefully interpreting problem statements to arrive at accurate solutions. By mastering these concepts, we can confidently tackle a wide range of mathematical challenges.

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