Inverse Functions If P(t) Is The Inverse Of T(p) And P(5) Equals 7

This statement delves into the fundamental concept of inverse functions and their relationship. To determine whether the statement is true or false, we need to dissect the definition of inverse functions and apply it to the given scenario. Let's embark on a journey to unravel this mathematical puzzle.

Grasping the Essence of Inverse Functions

At the heart of this problem lies the notion of inverse functions. In essence, an inverse function reverses the operation of the original function. Imagine a function as a machine that takes an input, processes it, and spits out an output. The inverse function acts as a machine that takes the output of the original function and transforms it back into the original input.

Mathematically, if we have a function $T(p)$ that maps an input $p$ to an output, say $T(p) = q$, then the inverse function, denoted as $P(t)$, would map the output $q$ back to the original input $p$. In other words, $P(q) = p$. This interplay between input and output is the crux of inverse functions. To truly grasp this concept, let's explore some concrete examples. Consider the function $T(p) = 2p$. This function doubles any input. Its inverse, $P(t) = t/2$, halves any input. If we input 3 into $T(p)$, we get $T(3) = 2 * 3 = 6$. Now, if we input 6 into $P(t)$, we get $P(6) = 6/2 = 3$, which is our original input. This exemplifies the reversing action of inverse functions. Another classic example is the relationship between exponential and logarithmic functions. The exponential function $T(p) = e^p$ raises the base e to the power of p. Its inverse, the natural logarithm $P(t) = ln(t)$, finds the power to which e must be raised to obtain t. If we compute $T(2) = e^2$, and then apply the inverse function $P(e^2) = ln(e^2)$, we get 2, illustrating the inverse relationship. Understanding this fundamental concept is vital for deciphering the given statement. The notation $P(t)$ being the inverse of $T(p)$ is crucial. It signifies that if $T$ takes an input $p$ and yields an output, say $q$, then $P$ should take that output $q$ as its input and return the original input $p$. This input-output reversal is the defining characteristic of inverse functions, and it's this characteristic that we'll leverage to analyze the statement.

Deconstructing the Given Information

Now, let's dissect the information presented in the statement. We are given that $P(t)$ is the inverse of $T(p)$. This is our key piece of information, as it establishes the fundamental relationship between the two functions. It tells us that if $T(p) = q$, then $P(q) = p$. This reversal of input and output is the essence of inverse functions, as we discussed earlier. We are also given that $P(5) = 7$. This is a specific data point that connects the input and output of the inverse function $P(t)$. It tells us that when the input to $P$ is 5, the output is 7. This information is crucial because it allows us to infer the corresponding relationship for the original function $T(p)$. Remember, inverse functions reverse the roles of input and output. So, if $P(5) = 7$, this implies a direct relationship for the original function $T(p)$. Specifically, it suggests that when the input to $T$ is 7, the output should be 5. This is because the inverse function $P$ takes 5 as input and produces 7 as output, meaning that $T$ must take 7 as input and produce 5 as output to maintain the inverse relationship. To solidify this understanding, let's consider an analogy. Imagine a lock and its key. The function $T$ is like locking the door with the key (input $p$) resulting in a locked door (output). The inverse function $P$ is like unlocking the door. It takes the locked door (input) and uses the key (output) to unlock it, reversing the process. In our case, $P(5) = 7$ is like saying that using a key (5) unlocks a specific lock (7). The statement is essentially asking if the reverse is true: if we use the lock (7), can we unlock it with the key (5)? The relationship between $P(5) = 7$ and the inverse function concept is the cornerstone of understanding this problem. It's from this specific data point that we can deduce the behavior of the original function $T(p)$.

Evaluating the Claim: T(7) = 5

The crux of the matter lies in evaluating the claim that $T(7) = 5$. To do this, we must rely on our understanding of inverse functions and the given information that $P(5) = 7$. Remember, if $P(t)$ is the inverse of $T(p)$, then $P(q) = p$ implies that $T(p) = q$. This interplay between input and output is the defining characteristic of inverse functions. We are given that $P(5) = 7$. This means that when we input 5 into the function $P$, the output is 7. Now, let's apply the principle of inverse functions. If $P(5) = 7$, then the original function $T$ must reverse this relationship. This implies that when we input 7 into the function $T$, the output must be 5. In other words, $T(7) = 5$. This is because the inverse function takes the output of the original function as its input and returns the original function's input. So, if $P$ takes 5 to 7, then $T$ must take 7 to 5. Let's solidify this with an example. Suppose $T(p) = 2p - 9$. To find its inverse, we can follow these steps:

1. Replace $T(p)$ with $t$: $t = 2p - 9$
2. Swap $t$ and $p$: $p = 2t - 9$
3. Solve for $t$: $p + 9 = 2t$, so $t = (p + 9)/2$
4. Replace $t$ with $P(p)$: $P(p) = (p + 9)/2$

Now, let's check if our given condition holds. If $P(5) = 7$, then $(5 + 9)/2 = 14/2 = 7$, which is true. Now, let's evaluate $T(7)$. $T(7) = 2(7) - 9 = 14 - 9 = 5$. Thus, $T(7) = 5$ holds true in this example. This concrete example reinforces the validity of the statement. The essence of inverse functions dictates that if $P(5) = 7$, then $T(7)$ must indeed equal 5. There is no other possibility if $P$ is truly the inverse of $T$. Therefore, the claim that $T(7) = 5$ is logically sound and consistent with the definition of inverse functions. In conclusion, the statement is demonstrably true.

Conclusion: The Verdict

After a thorough examination of the concept of inverse functions and the given information, we can confidently conclude that the statement is True. The fundamental property of inverse functions, which is the reversal of input and output, dictates that if $P(5) = 7$ and $P(t)$ is the inverse of $T(p)$, then $T(7)$ must indeed equal 5. This conclusion is not just a matter of speculation; it is a direct consequence of the definition and behavior of inverse functions. Our analysis has shown that the provided information logically leads to the stated outcome. By understanding the interplay between a function and its inverse, we can navigate such problems with clarity and precision. This exercise highlights the importance of grasping fundamental mathematical concepts, as they form the bedrock upon which more complex ideas are built.

Answer: A. True