Function Domain And Range: Could This Be True?

Hey guys, let's dive into a super interesting math problem that's all about understanding functions, their domains, and their ranges. We've got a function, let's call it $g$, and we're given some pretty specific details about it. The domain of $g$ is between $-20$ and $5$, inclusive. This means the possible $x$-values for our function are all the numbers from $-20$ up to $5$. Mathematically, we write this as -20 r_eq x r_eq 5. The range of $g$ is even more constrained; it's between $-5$ and $45$, inclusive. This tells us that the possible output values, the $g(x)$ values, are all the numbers from $-5$ up to $45$. So, -5 r_eq g(x) r_eq 45.

Now, we're also given two specific points on this function: $g(0) = -2$ and $g(-9) = 6$. These are like little hints that tell us exactly what the function does at certain input values. The point $(0, -2)$ is definitely on the graph of $g$, and so is the point $(-9, 6)$. Both of these points must satisfy the domain and range restrictions we just talked about. Let's quickly check: for $(0, -2)$, $0$ is within $[-20, 5]$ and $-2$ is within $[-5, 45]$. Perfect. For $(-9, 6)$, $-9$ is within $[-20, 5]$ and $6$ is within $[-5, 45]$. Everything checks out so far!

Our mission, should we choose to accept it, is to figure out which of the given statements could be true for this function $g$. This means we need to examine each option and see if it violates any of the rules we've established for $g$. Remember, a function assigns exactly one output for each input, and in this case, all inputs and outputs must be within the specified domain and range. Let's put on our detective hats and analyze each clue. We're looking for a statement that doesn't break any of the rules. It's like a puzzle, and we need to find the piece that fits perfectly without any force.

We're given four possibilities, and only one of them is a valid scenario for our function $g$. So, we'll go through them one by one. Think of it like checking if a potential suspect was actually at the scene of the crime. If they have an alibi (meaning they were somewhere else), then they couldn't have done it. Similarly, if a potential point for $g$ falls outside the domain or range, it simply cannot be true for this function. It's a process of elimination, really. We want to find the statement that is consistent with all the information we have about $g$. This involves carefully checking both the input value (the $x$-value) against the domain and the output value (the $g(x)$ value) against the range. Don't forget that we also know two specific points that are on the function, and these can sometimes help us reason about the function's behavior, although for this particular question, the domain and range constraints will be our primary tools.

Let's get started with option A. This is where the real work begins, and we'll see which of these statements is the one that $g$ could possibly satisfy. It's all about making sure that for any proposed point $(x, g(x))$, the $x$ is in the domain $[-20, 5]$ and $g(x)$ is in the range $[-5, 45]$. Any violation means that statement is a no-go. We're aiming for that one correct choice that respects all the boundaries. So, let's scrutinize each one carefully, shall we? The journey to the correct answer starts now!

Analyzing the Options

Alright guys, let's break down each of the given statements to see which one could be true for our function $g$. Remember, the key rules are: the domain is -20 r_eq x r_eq 5, and the range is -5 r_eq g(x) r_eq 45. We also know for sure that $g(0)=-2$ and $g(-9)=6$. Let's examine each option with these constraints in mind. It’s all about making sure the proposed input ($x$) is within the domain and the proposed output ($g(x)$) is within the range.

Option A: $g(-4)=-11$

First up, we have $g(-4)=-11$. We need to check two things here: is the input, $-4$, within the domain, and is the output, $-11$, within the range? Let's check the input: is $-4$ between $-20$ and $5$? Yes, -20 r_eq -4 r_eq 5. So far, so good. Now let's check the output: is $-11$ between $-5$ and $45$? Uh oh. $-11$ is less than $-5$. This means that $-11$ is outside the allowed range of the function $g$. Since the range dictates all possible output values, $g(x)$ can never be $-11$. Therefore, the statement $g(-4)=-11$ cannot be true for function $g$. This option is out, folks.

Option B: $g(0)=2$

Next, let's look at option B: $g(0)=2$. We know the domain is -20 r_eq x r_eq 5, and the range is -5 r_eq g(x) r_eq 45. Let's check the input: is $0$ within the domain? Yes, -20 r_eq 0 r_eq 5. So, the input is valid. Now let's check the output: is $2$ within the range? Yes, -5 r_eq 2 r_eq 45. The output is also valid. So, based on domain and range, this could be true. However, remember we were given a very specific piece of information: $g(0)=-2$. This means that when the input is $0$, the output must be $-2$. It cannot be $2$. A function can only have one output for a given input. Since we are explicitly told $g(0)=-2$, it's impossible for $g(0)$ to also equal $2$. This statement contradicts a given fact about the function. So, option B cannot be true either. It’s a bummer, but it’s just not possible given the initial conditions.

Option C: $g(7)=-1$

Moving on to option C: $g(7)=-1$. Let's check the input first. Is $7$ within the domain -20 r_eq x r_eq 5? No, $7$ is greater than $5$. This means $7$ is outside the domain of the function $g$. The function $g$ is not defined for an input of $7$. Therefore, the statement $g(7)=-1$ cannot be true for function $g$. This one is also eliminated. We're narrowing it down, aren't we?

Option D: $g(-13)=20$

Finally, let's examine option D: $g(-13)=20$. We need to check if the input $-13$ is in the domain and if the output $20$ is in the range. First, the input: is $-13$ within the domain -20 r_eq x r_eq 5? Yes, $-13$ is indeed between $-20$ and $5$. So, the input is valid. Now, let's check the output: is $20$ within the range -5 r_eq g(x) r_eq 45? Yes, $20$ is between $-5$ and $45$. So, the output is also valid.

Crucially, this statement $g(-13)=20$ does not contradict any of the given information. It doesn't violate the domain, it doesn't violate the range, and it doesn't contradict the specific points we know, $g(0)=-2$ and $g(-9)=6$. It's entirely possible for a function with the given domain and range to have the point $(-13, 20)$ on its graph. There are no restrictions that prevent this from being true. Therefore, the statement $g(-13)=20$ could be true for function $g$.

Conclusion

So, after carefully analyzing all four options, we've found that options A, B, and C all violate at least one of the conditions set for the function $g$. Option A failed because the output value was outside the range. Option B failed because it contradicted a specific given point of the function, $g(0)=-2$. Option C failed because the input value was outside the domain.

This leaves us with Option D: $g(-13)=20$ as the only statement that could be true. The input $-13$ is within the domain $[-20, 5]$, and the output $20$ is within the range $[-5, 45]$. This point is consistent with all the information provided about function $g$. It doesn't break any rules, and it doesn't contradict any known facts. So, when you're tackling these kinds of problems, always remember to check both the domain for the input and the range for the output. And don't forget any specific points given – they are absolute truths for that function! Keep practicing, and you'll become a function master in no time. Stay curious, mathletes!