Answer :
Let's solve the problem step-by-step to determine the domain and range of the function [tex]\( G(x) = 6500 \sqrt{x^2 + 4} \)[/tex].
### Finding the Domain
The domain of a function is the set of all possible values of the independent variable (in this case, [tex]\( x \)[/tex]) for which the function is defined.
1. Identify the variable: Here, [tex]\( x \)[/tex] represents the length of the cable in meters.
2. Consider the mathematical expression inside the function: The expression [tex]\( x^2 + 4 \)[/tex] appears inside a square root. The square root function [tex]\( \sqrt{y} \)[/tex] is defined for all [tex]\( y \geq 0 \)[/tex].
3. Determine the conditions for [tex]\( x \)[/tex]: Since [tex]\( x^2 + 4 \)[/tex] will always be greater than or equal to 4 (because [tex]\( x^2 \)[/tex] is always non-negative and 4 is added to it),
- [tex]\( x^2 + 4 \geq 4 \)[/tex]
- This means the expression inside the square root is always non-negative, regardless of the value of [tex]\( x \)[/tex].
4. Determine the realistic values of [tex]\( x \)[/tex]: In the context of the problem, [tex]\( x \)[/tex] represents a physical length (in meters), so it cannot be negative.
Therefore, [tex]\( x \)[/tex] can be any non-negative real number.
Thus, the domain of the function is:
[tex]\[ x \geq 0 \][/tex]
### Finding the Range
The range of a function is the set of all possible values of the function (in this case, [tex]\( G(x) \)[/tex]).
1. Evaluate [tex]\( G(x) \)[/tex]: The function given is [tex]\( G(x) = 6500 \sqrt{x^2 + 4} \)[/tex].
2. Find the minimum value of [tex]\( G(x) \)[/tex]: To find the minimum value, we need to consider the smallest value [tex]\( x \)[/tex] can take within its domain.
- When [tex]\( x = 0 \)[/tex], the value of the function is:
[tex]\[ G(0) = 6500 \sqrt{0^2 + 4} = 6500 \sqrt{4} = 6500 \times 2 = 13000 \][/tex]
3. Behavior as [tex]\( x \)[/tex] increases: As [tex]\( x \)[/tex] increases, [tex]\( x^2 + 4 \)[/tex] increases, and so does [tex]\( \sqrt{x^2 + 4} \)[/tex]. Hence, [tex]\( G(x) \)[/tex] will increase without bound as [tex]\( x \)[/tex] increases.
- Therefore, [tex]\( G(x) \)[/tex] only gets larger from 13000 upwards.
4. Summarize the range: Since [tex]\( G(x) \)[/tex] starts at 13000 when [tex]\( x = 0 \)[/tex] and increases indefinitely as [tex]\( x \)[/tex] increases, the range of the function is all values [tex]\( y \)[/tex] such that [tex]\( y \geq 13000 \)[/tex].
Thus, the range of the function is:
[tex]\[ y \geq 13000 \][/tex]
### Conclusion
The domain of the function [tex]\( G(x) = 6500 \sqrt{x^2 + 4} \)[/tex] is:
[tex]\[ x \geq 0 \][/tex]
The range of the function is:
[tex]\[ y \geq 13000 \][/tex]
### Finding the Domain
The domain of a function is the set of all possible values of the independent variable (in this case, [tex]\( x \)[/tex]) for which the function is defined.
1. Identify the variable: Here, [tex]\( x \)[/tex] represents the length of the cable in meters.
2. Consider the mathematical expression inside the function: The expression [tex]\( x^2 + 4 \)[/tex] appears inside a square root. The square root function [tex]\( \sqrt{y} \)[/tex] is defined for all [tex]\( y \geq 0 \)[/tex].
3. Determine the conditions for [tex]\( x \)[/tex]: Since [tex]\( x^2 + 4 \)[/tex] will always be greater than or equal to 4 (because [tex]\( x^2 \)[/tex] is always non-negative and 4 is added to it),
- [tex]\( x^2 + 4 \geq 4 \)[/tex]
- This means the expression inside the square root is always non-negative, regardless of the value of [tex]\( x \)[/tex].
4. Determine the realistic values of [tex]\( x \)[/tex]: In the context of the problem, [tex]\( x \)[/tex] represents a physical length (in meters), so it cannot be negative.
Therefore, [tex]\( x \)[/tex] can be any non-negative real number.
Thus, the domain of the function is:
[tex]\[ x \geq 0 \][/tex]
### Finding the Range
The range of a function is the set of all possible values of the function (in this case, [tex]\( G(x) \)[/tex]).
1. Evaluate [tex]\( G(x) \)[/tex]: The function given is [tex]\( G(x) = 6500 \sqrt{x^2 + 4} \)[/tex].
2. Find the minimum value of [tex]\( G(x) \)[/tex]: To find the minimum value, we need to consider the smallest value [tex]\( x \)[/tex] can take within its domain.
- When [tex]\( x = 0 \)[/tex], the value of the function is:
[tex]\[ G(0) = 6500 \sqrt{0^2 + 4} = 6500 \sqrt{4} = 6500 \times 2 = 13000 \][/tex]
3. Behavior as [tex]\( x \)[/tex] increases: As [tex]\( x \)[/tex] increases, [tex]\( x^2 + 4 \)[/tex] increases, and so does [tex]\( \sqrt{x^2 + 4} \)[/tex]. Hence, [tex]\( G(x) \)[/tex] will increase without bound as [tex]\( x \)[/tex] increases.
- Therefore, [tex]\( G(x) \)[/tex] only gets larger from 13000 upwards.
4. Summarize the range: Since [tex]\( G(x) \)[/tex] starts at 13000 when [tex]\( x = 0 \)[/tex] and increases indefinitely as [tex]\( x \)[/tex] increases, the range of the function is all values [tex]\( y \)[/tex] such that [tex]\( y \geq 13000 \)[/tex].
Thus, the range of the function is:
[tex]\[ y \geq 13000 \][/tex]
### Conclusion
The domain of the function [tex]\( G(x) = 6500 \sqrt{x^2 + 4} \)[/tex] is:
[tex]\[ x \geq 0 \][/tex]
The range of the function is:
[tex]\[ y \geq 13000 \][/tex]
