Finding The Inverse Of $f(x)=9x^2-12$ A Step-by-Step Guide
In the realm of mathematics, the concept of an inverse function holds significant importance. When we are given a function, the inverse function essentially "undoes" the original function. This means that if we apply a function to a value and then apply its inverse to the result, we end up with the original value. This exploration delves into the process of finding the inverse of a function, with a specific focus on the function $f(x) = 9x^2 - 12$ where $x extgreater= 0$. To properly understand this, let's define what an inverse function is in more detail.
Understanding Inverse Functions
An inverse function is a function that reverses the effect of the original function. If the original function $f$ maps an input $x$ to an output $y$, then the inverse function, denoted as $f^{-1}$, maps $y$ back to $x$. Not all functions have inverses; a function must be bijective (both injective and surjective) to have an inverse. An injective function (or one-to-one function) ensures that each input maps to a unique output, while a surjective function (or onto function) ensures that every possible output is mapped to by some input.
The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For a function to have an inverse, it must pass the horizontal line test, which is a visual way to determine if a function is one-to-one. If any horizontal line intersects the graph of the function more than once, the function does not have an inverse.
Steps to Find the Inverse Function
To find the inverse of a function, we typically follow these steps:
- Replace $f(x)$ with $y$. This makes the equation easier to manipulate.
- Swap $x$ and $y$. This is the key step in finding the inverse, as it reverses the roles of input and output.
- Solve for $y$. This expresses the inverse function in terms of $x$.
- Replace $y$ with $f^{-1}(x)$. This is the standard notation for the inverse function.
It's crucial to consider the domain restriction, as this often plays a critical role in determining the correct inverse. In our case, the domain of $f(x)$ is restricted to $x extgreater= 0$, which means we only consider non-negative values of $x$. This restriction is essential for the function to have a well-defined inverse.
Applying the Steps to $f(x) = 9x^2 - 12, x \textgreater= 0$
Let's apply these steps to the given function $f(x) = 9x^2 - 12$, where $x extgreater= 0$:
- Replace $f(x)$ with $y$: $y = 9x^2 - 12$
- Swap $x$ and $y$: $x = 9y^2 - 12$
- Solve for $y$:
- Add 12 to both sides: $x + 12 = 9y^2$
- Divide both sides by 9: $\frac{x + 12}{9} = y^2$
- Take the square root of both sides: $y = $
- Simplify the square root: $y = \pm\frac{\sqrt{x + 12}}{3}$
- Consider the domain restriction: Since the original function is defined for $x \textgreater= 0$, we need to determine the appropriate sign for the square root. When finding the inverse, we look at the range of the original function. For $f(x) = 9x^2 - 12$, when $x \textgreater= 0$, the range is $y \textgreater= -12$. The domain of the inverse function will be the range of the original function, and the range of the inverse function will be the domain of the original function. Since the domain of the original function is $x \textgreater= 0$, we choose the positive square root.
- Replace $y$ with $f^-1}(x)$(x) = \frac{\sqrt{x + 12}}{3}$
Thus, the inverse of the function $f(x) = 9x^2 - 12$, where $x \textgreater= 0$, is $f^{-1}(x) = \frac{\sqrt{x + 12}}{3}$. This methodical approach ensures that we correctly identify the inverse function, taking into account any domain restrictions that might be present.
When seeking the inverse of the function $f(x) = 9x^2 - 12$ for $x \geq 0$, we derived the inverse function as $f^{-1}(x) = \frac{\sqrt{x + 12}}{3}$. Now, let's examine the given options to determine which one matches our result and understand why the others might be incorrect. This detailed comparison helps in reinforcing the understanding of inverse functions and the importance of following the correct steps in their derivation. The focus here is on identifying the option that is mathematically equivalent to our derived inverse function and also consistent with the domain and range considerations. This analytical step is crucial in problem-solving and ensuring the accuracy of the final answer.
Option A: $h(x) = \frac{\sqrt{x - 12}}{3}$
Let's analyze Option A, which states that the inverse function is $h(x) = \frac{\sqrt{x - 12}}{3}$. This option looks similar to our derived inverse function, but there's a crucial difference under the square root. Instead of $x + 12$, we have $x - 12$. To understand why this is incorrect, let's trace back the steps we took to find the inverse.
When we swapped $x$ and $y$ in the original equation $y = 9x^2 - 12$, we obtained $x = 9y^2 - 12$. The next step involved isolating the term with $y$, which meant adding 12 to both sides, resulting in $x + 12 = 9y^2$. It is this $x + 12$ that is under the square root in the correct inverse function. Option A incorrectly subtracts 12 from $x$ under the square root, indicating a flaw in the algebraic manipulation.
Furthermore, let's consider the domain of this function. For $h(x) = \frac{\sqrt{x - 12}}{3}$ to be defined, the expression inside the square root, $x - 12$, must be non-negative. This means $x - 12 \geq 0$, which implies $x \geq 12$. The domain of this proposed inverse function is $x \geq 12$. However, the range of the original function $f(x) = 9x^2 - 12$ for $x \geq 0$ is $y \geq -12$. Therefore, the domain of the inverse function should be $x \geq -12$, not $x \geq 12$. This discrepancy in the domain further confirms that Option A is not the correct inverse function.
In summary, the subtraction of 12 under the square root and the incorrect domain make Option A an invalid choice for the inverse of $f(x) = 9x^2 - 12$ for $x \geq 0$. The correct inverse function must account for adding 12 when isolating the $y^2$ term and must have a domain that aligns with the range of the original function.
Correct Answer
After carefully deriving the inverse function and analyzing the options, we find that the correct inverse of $f(x) = 9x^2 - 12$ where $x \geq 0$ is:
This function is obtained by swapping $x$ and $y$ in the original equation, solving for $y$, and considering the domain restriction. The domain of the original function is $x \geq 0$, and its range is $y \geq -12$. Therefore, the domain of the inverse function is $x \geq -12$, and its range is $y \geq 0$, which aligns with the original function's domain.
This detailed analysis ensures a comprehensive understanding of how to find the inverse of a function and the importance of considering domain restrictions during the process. By following these steps and carefully evaluating the options, we can confidently identify the correct inverse function.
In conclusion, determining the inverse of a function requires a systematic approach, paying close attention to algebraic manipulations and domain restrictions. For the function $f(x) = 9x^2 - 12$, with the domain restricted to $x \geq 0$, the inverse function is found to be $f^{-1}(x) = \frac{\sqrt{x + 12}}{3}$. This result is achieved by swapping the variables $x$ and $y$, solving for $y$, and considering the domain of the original function to ensure the appropriate sign is chosen for the square root. By carefully following these steps and understanding the fundamental principles of inverse functions, one can confidently tackle such problems. This process not only yields the correct answer but also enhances the understanding of the relationship between a function and its inverse, a critical concept in mathematics. The analysis also highlighted the importance of carefully examining each step in the derivation to avoid common pitfalls, such as incorrect algebraic manipulations or overlooking domain restrictions, which can lead to incorrect solutions. The ability to accurately find and interpret inverse functions is a valuable skill in various mathematical contexts, making a thorough understanding of this topic essential.
