Function F(x) Ordered Pairs: True Statement?

Hey guys, let's dive into the world of functions and ordered pairs today. We've got a specific function, $f(x)$, defined by a set of ordered pairs: $\{(8,-3),(0,4),(1,-5),(2,-1),(-6,10)\}$. Our mission, should we choose to accept it, is to figure out which statement among the given options is actually true about this function. This is all about understanding what ordered pairs represent in the context of a function. Remember, an ordered pair $(x, y)$ means that when the input is $x$, the output is $y$. So, $f(x) = y$. Let's break down each option and see if it aligns with the definition of our function. We'll be looking for a direct match within the given set of pairs. It's like a matching game, but with math! So, pay close attention to the inputs and outputs. The function is essentially a machine: you put in an $x$ value, and it spits out a specific $y$ value. For a function to be valid, each input $x$ can only have one output $y$. Our given set of pairs already satisfies this condition, which is great. Now, we just need to test the specific claims made in options A, B, C, and D against these pairs. This requires careful observation and a solid grasp of function notation. Don't rush through it; take your time to match the $x$ and $y$ values correctly. We're not just looking for any relationship; we're looking for the exact relationship as defined by the ordered pairs. So, when we see something like $f(-3)=8$, we need to ask ourselves: is there an ordered pair in our set that looks like $(-3, 8)$? If not, that statement is false. Let's get ready to put our detective hats on and solve this math mystery!

Understanding Ordered Pairs and Function Notation

Alright, let's get a bit more detailed about these ordered pairs and what they mean for our function $f(x)$. When we see a set of ordered pairs like $\{(8,-3),(0,4),(1,-5),(2,-1),(-6,10)\}$, we're essentially looking at a mapping. The first number in each pair is the input, often called the domain value, and it's the $x$ in $f(x)$. The second number is the output, which is part of the range, and it's the $y$ in $f(x)$. So, for instance, the pair $(8, -3)$ tells us that $f(8) = -3$. This means when you plug in $8$ into the function $f$, the result you get is $-3$. Similarly, $(0, 4)$ means $f(0) = 4$, $(1, -5)$ means $f(1) = -5$, $(2, -1)$ means $f(2) = -1$, and finally, $(-6, 10)$ means $f(-6) = 10$. It's crucial to remember that for any given input $x$, there can only be one output $y$. This is the fundamental rule of functions. If we had, say, $(8, -3)$ and $(8, 5)$ in the same set, it wouldn't represent a function because the input $8$ would have two different outputs. Thankfully, our provided set adheres to this rule, so we're dealing with a legitimate function.

Now, let's tackle the options provided. We need to check each one against our defined function. This means we're looking for a specific ordered pair within our set that matches the statement.

Analyzing the Options

Option A: $f(-3)=8$

To verify this statement, we need to see if there's an ordered pair in our set where the input (the first number) is $-3$ and the output (the second number) is $8$. In other words, we're looking for the ordered pair $(-3, 8)$. Let's scan our set: $\{(8,-3),(0,4),(1,-5),(2,-1),(-6,10)\}$. Do we see $(-3, 8)$ in there? Nope. We have $(8, -3)$, which means $f(8) = -3$, but that's not the same as $f(-3) = 8$. So, statement A is false. Remember, the order matters in an ordered pair!

Option B: $f(3)=5$

Similar to option A, we need to check if the ordered pair $(3, 5)$ exists in our function set. Let's look again: $\{(8,-3),(0,4),(1,-5),(2,-1),(-6,10)\}$. Is there a pair starting with $3$ and ending with $5$? No, there isn't. We have $(1, -5)$, which means $f(1) = -5$, but this is not what statement B claims. Therefore, statement B is also false.

Option C: $f(8)=0$

Here, we're checking if the input $8$ gives an output of $0$. This translates to looking for the ordered pair $(8, 0)$ in our set. Let's examine our function: $\{(8,-3),(0,4),(1,-5),(2,-1),(-6,10)\}$. We do see an ordered pair that starts with $8$, it's $(8, -3)$. This means $f(8) = -3$. Since the output is $-3$ and not $0$, statement C is false.

Option D: $f(-6)=10$

Finally, let's check this last statement. We need to see if the input $-6$ yields an output of $10$. This means we are searching for the ordered pair $(-6, 10)$ within our given set. Looking at $\{(8,-3),(0,4),(1,-5),(2,-1),(-6,10)\}$, we can clearly spot the ordered pair $(-6, 10)$. This pair directly tells us that when the input is $-6$, the output is $10$. Thus, $f(-6) = 10$. This statement perfectly matches one of the ordered pairs defining the function. Therefore, statement D is true.

Conclusion: The True Statement

After meticulously examining each option against the given set of ordered pairs for the function $f(x)$, we've concluded that only one statement holds true. The function is defined by $\{(8,-3),(0,4),(1,-5),(2,-1),(-6,10)\}$. This means:

• $f(8) = -3$
• $f(0) = 4$
• $f(1) = -5$
• $f(2) = -1$
• $f(-6) = 10$

Let's revisit our options:

• A. $f(-3)=8$ - False, as $(-3, 8)$ is not in the set.
• B. $f(3)=5$ - False, as $(3, 5)$ is not in the set.
• C. $f(8)=0$ - False, as $f(8)$ is $-3$, not $0$.
• D. $f(-6)=10$ - True, as the ordered pair $(-6, 10)$ is present in the set.

So, the correct statement regarding the function $f(x)$ is D. $f(-6)=10$. It's all about matching those inputs to their corresponding outputs as defined by the pairs. Great job working through this, guys! Keep practicing, and these function concepts will become second nature.