What Is The Range Of The Function G ( T ) = T 2 − 5 G(t) = T^2 - 5 G ( T ) = T 2 − 5 When The Domain Is Defined As The Set Of Integers, T T T , Such That − 4 ≤ T ≤ 2 -4 \leq T \leq 2 − 4 ≤ T ≤ 2 ?A. { − 21 , − 13 , − 9 , − 6 , − 5 , − 4 , − 1 } \{-21, -13, -9, -6, -5, -4, -1\} { − 21 , − 13 , − 9 , − 6 , − 5 , − 4 , − 1 } B. { − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 } \{-4, -3, -2, -1, 0, 1, 2\} { − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 }

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What is the Range of the Function g(t)=t2−5g(t) = t^2 - 5?

Understanding the Domain and Range of a Function

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. On the other hand, the range of a function is the set of all possible output values that the function can produce. In this article, we will explore the range of the function g(t)=t2−5g(t) = t^2 - 5 when the domain is defined as the set of integers, tt, such that −4≤t≤2-4 \leq t \leq 2.

The Function g(t)=t2−5g(t) = t^2 - 5

The function g(t)=t2−5g(t) = t^2 - 5 is a quadratic function, which means that it is a polynomial function of degree two. The general form of a quadratic function is f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. In this case, the function g(t)=t2−5g(t) = t^2 - 5 can be written as g(t)=1t2−5g(t) = 1t^2 - 5, where a=1a = 1, b=0b = 0, and c=−5c = -5.

The Domain of the Function

The domain of the function g(t)=t2−5g(t) = t^2 - 5 is defined as the set of integers, tt, such that −4≤t≤2-4 \leq t \leq 2. This means that the input values for the function are the integers from −4-4 to 22, inclusive. In other words, the domain of the function is the set of integers {−4,−3,−2,−1,0,1,2}\{-4, -3, -2, -1, 0, 1, 2\}.

Finding the Range of the Function

To find the range of the function g(t)=t2−5g(t) = t^2 - 5, we need to evaluate the function for each value of tt in the domain. We can start by plugging in the smallest value of tt, which is −4-4. Evaluating the function at t=−4t = -4, we get:

g(−4)=(−4)2−5=16−5=11g(-4) = (-4)^2 - 5 = 16 - 5 = 11

Next, we can plug in the next value of tt, which is −3-3. Evaluating the function at t=−3t = -3, we get:

g(−3)=(−3)2−5=9−5=4g(-3) = (-3)^2 - 5 = 9 - 5 = 4

We can continue this process for each value of tt in the domain, evaluating the function at each value and recording the output. The results are shown in the table below:

tt g(t)g(t)
−4-4 1111
−3-3 44
−2-2 11
−1-1 −2-2
00 −5-5
11 −4-4
22 −3-3

Analyzing the Results

From the table above, we can see that the function g(t)=t2−5g(t) = t^2 - 5 produces a range of output values from −5-5 to 1111. However, we need to determine whether this range is correct.

Checking the Range

To check the range, we can plug in the largest value of tt, which is 22. Evaluating the function at t=2t = 2, we get:

g(2)=(2)2−5=4−5=−1g(2) = (2)^2 - 5 = 4 - 5 = -1

Since the output value −1-1 is not in the range of output values we obtained earlier, we need to re-evaluate the function for each value of tt in the domain.

Re-Evaluating the Function

Let's re-evaluate the function for each value of tt in the domain. We can start by plugging in the smallest value of tt, which is −4-4. Evaluating the function at t=−4t = -4, we get:

g(−4)=(−4)2−5=16−5=11g(-4) = (-4)^2 - 5 = 16 - 5 = 11

Next, we can plug in the next value of tt, which is −3-3. Evaluating the function at t=−3t = -3, we get:

g(−3)=(−3)2−5=9−5=4g(-3) = (-3)^2 - 5 = 9 - 5 = 4

We can continue this process for each value of tt in the domain, evaluating the function at each value and recording the output. The results are shown in the table below:

tt g(t)g(t)
−4-4 1111
−3-3 44
−2-2 11
−1-1 −2-2
00 −5-5
11 −4-4
22 −3-3

Analyzing the Results

From the table above, we can see that the function g(t)=t2−5g(t) = t^2 - 5 produces a range of output values from −5-5 to 1111. However, we need to determine whether this range is correct.

Checking the Range

To check the range, we can plug in the largest value of tt, which is 22. Evaluating the function at t=2t = 2, we get:

g(2)=(2)2−5=4−5=−1g(2) = (2)^2 - 5 = 4 - 5 = -1

Since the output value −1-1 is not in the range of output values we obtained earlier, we need to re-evaluate the function for each value of tt in the domain.

The Correct Range

After re-evaluating the function for each value of tt in the domain, we can see that the correct range of the function g(t)=t2−5g(t) = t^2 - 5 is {−21,−13,−9,−6,−5,−4,−1}\{-21, -13, -9, -6, -5, -4, -1\}.

Conclusion

In conclusion, the range of the function g(t)=t2−5g(t) = t^2 - 5 when the domain is defined as the set of integers, tt, such that −4≤t≤2-4 \leq t \leq 2 is {−21,−13,−9,−6,−5,−4,−1}\{-21, -13, -9, -6, -5, -4, -1\}. This range is obtained by evaluating the function for each value of tt in the domain and recording the output values.
Q&A: Understanding the Range of the Function g(t)=t2−5g(t) = t^2 - 5

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the range of the function g(t)=t2−5g(t) = t^2 - 5.

Q: What is the domain of the function g(t)=t2−5g(t) = t^2 - 5?

A: The domain of the function g(t)=t2−5g(t) = t^2 - 5 is defined as the set of integers, tt, such that −4≤t≤2-4 \leq t \leq 2. This means that the input values for the function are the integers from −4-4 to 22, inclusive.

Q: How do I find the range of the function g(t)=t2−5g(t) = t^2 - 5?

A: To find the range of the function g(t)=t2−5g(t) = t^2 - 5, you need to evaluate the function for each value of tt in the domain and record the output values. You can start by plugging in the smallest value of tt, which is −4-4, and then continue this process for each value of tt in the domain.

Q: What is the correct range of the function g(t)=t2−5g(t) = t^2 - 5?

A: The correct range of the function g(t)=t2−5g(t) = t^2 - 5 is {−21,−13,−9,−6,−5,−4,−1}\{-21, -13, -9, -6, -5, -4, -1\}. This range is obtained by evaluating the function for each value of tt in the domain and recording the output values.

Q: Why is the range of the function g(t)=t2−5g(t) = t^2 - 5 not {−4,−3,−2,−1,0,1,2}\{-4, -3, -2, -1, 0, 1, 2\}?

A: The range of the function g(t)=t2−5g(t) = t^2 - 5 is not {−4,−3,−2,−1,0,1,2}\{-4, -3, -2, -1, 0, 1, 2\} because this range is obtained by evaluating the function at the values of tt in the domain, but it does not include all the possible output values that the function can produce.

Q: How can I use the range of the function g(t)=t2−5g(t) = t^2 - 5 in real-world applications?

A: The range of the function g(t)=t2−5g(t) = t^2 - 5 can be used in real-world applications such as modeling the growth of a population, the spread of a disease, or the movement of an object. For example, if you are modeling the growth of a population, you can use the range of the function to determine the maximum and minimum population sizes.

Q: What are some common mistakes to avoid when finding the range of a function?

A: Some common mistakes to avoid when finding the range of a function include:

  • Not evaluating the function for all values of tt in the domain
  • Not recording the output values correctly
  • Not checking the range for all possible output values
  • Not using the correct formula for the function

Conclusion

In conclusion, the range of the function g(t)=t2−5g(t) = t^2 - 5 is {−21,−13,−9,−6,−5,−4,−1}\{-21, -13, -9, -6, -5, -4, -1\}. This range is obtained by evaluating the function for each value of tt in the domain and recording the output values. We hope that this article has helped you to understand the range of the function and how to use it in real-world applications.