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ChatGPT
Aug 25
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, the function is y = 4/(x-5) - 3.
The function is defined for all real numbers except for x = 5, because dividing by zero is undefined. Therefore, the domain of the function is all real numbers except x = 5.
The range of a function is the set of all possible output values (y-values) that the function can take. In this case, the function is y = 4/(x-5) - 3.
As x approaches 5 from either side, the value of the function approaches negative infinity. Therefore, the range of the function is all real numbers except negative infinity.
The function is defined for all real numbers except for x = 5, because dividing by zero is undefined. Therefore, the domain of the function is all real numbers except x = 5.
The range of a function is the set of all possible output values (y-values) that the function can take. In this case, the function is y = 4/(x-5) - 3.
As x approaches 5 from either side, the value of the function approaches negative infinity. Therefore, the range of the function is all real numbers except negative infinity.
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Google Gemini AI
Aug 25
Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. The function y = 4/(x-5) - 3 is defined for all real numbers except x = 5. This is because division by zero is undefined, and x - 5 would be zero when x = 5. Therefore, the domain of y = 4/(x-5) - 3 is all real numbers except x = 5. In interval notation, the domain can be expressed as:
```
Domain: (-∞, 5) U (5, ∞)
```
Range
The range of a function is the set of all possible output values (y-values) that the function can produce. To find the range of y = 4/(x-5) - 3, we can analyze the function's behavior as x approaches different values.
As x approaches infinity (either positive or negative), the term (x-5) becomes very large, and the fraction 4/(x-5) approaches zero. Therefore, y approaches -3 as x approaches infinity.
As x approaches 5 from the left (x < 5), the term (x-5) approaches zero from the negative side. This makes the fraction 4/(x-5) very large and negative, and y approaches negative infinity.
As x approaches 5 from the right (x > 5), the term (x-5) approaches zero from the positive side. This makes the fraction 4/(x-5) very large and positive, and y approaches positive infinity.
Based on this analysis, the range of y = 4/(x-5) - 3 is all real numbers except y = -3. In interval notation, the range can be expressed as:
```
Range: (-∞, -3) U (-3, ∞)
```
The domain of a function is the set of all possible input values (x-values) for which the function is defined. The function y = 4/(x-5) - 3 is defined for all real numbers except x = 5. This is because division by zero is undefined, and x - 5 would be zero when x = 5. Therefore, the domain of y = 4/(x-5) - 3 is all real numbers except x = 5. In interval notation, the domain can be expressed as:
```
Domain: (-∞, 5) U (5, ∞)
```
Range
The range of a function is the set of all possible output values (y-values) that the function can produce. To find the range of y = 4/(x-5) - 3, we can analyze the function's behavior as x approaches different values.
As x approaches infinity (either positive or negative), the term (x-5) becomes very large, and the fraction 4/(x-5) approaches zero. Therefore, y approaches -3 as x approaches infinity.
As x approaches 5 from the left (x < 5), the term (x-5) approaches zero from the negative side. This makes the fraction 4/(x-5) very large and negative, and y approaches negative infinity.
As x approaches 5 from the right (x > 5), the term (x-5) approaches zero from the positive side. This makes the fraction 4/(x-5) very large and positive, and y approaches positive infinity.
Based on this analysis, the range of y = 4/(x-5) - 3 is all real numbers except y = -3. In interval notation, the range can be expressed as:
```
Range: (-∞, -3) U (-3, ∞)
```
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