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To accurately compare the domain and range of the functions [tex]\( f(x) = 3x^2 \)[/tex], [tex]\( g(x) = \frac{1}{3x} \)[/tex], and [tex]\( h(x) = 3x \)[/tex], we need to analyze each function individually.
### Function [tex]\( f(x) = 3x^2 \)[/tex]
1. Domain: The function [tex]\( f(x) = 3x^2 \)[/tex] is a polynomial, which means it is defined for all real numbers. Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
2. Range: Since [tex]\( f(x) = 3x^2 \)[/tex] is a quadratic function that opens upwards (as the coefficient of [tex]\( x^2 \)[/tex] is positive), the minimum value of [tex]\( f(x) \)[/tex] is 0 (when [tex]\( x = 0 \)[/tex]). Thus, the range of [tex]\( f(x) \)[/tex] is all non-negative real numbers (i.e., [tex]\( [0, \infty) \)[/tex]).
### Function [tex]\( g(x) = \frac{1}{3x} \)[/tex]
1. Domain: The function [tex]\( g(x) = \frac{1}{3x} \)[/tex] involves division by [tex]\( x \)[/tex]. Division by zero is undefined, so [tex]\( x \)[/tex] cannot be 0. Therefore, the domain of [tex]\( g(x) \)[/tex] is all real numbers except 0.
2. Range: Since [tex]\( g(x) = \frac{1}{3x} \)[/tex] can never be 0 (it approaches infinity or negative infinity as [tex]\( x \)[/tex] gets close to 0, and it takes all positive and negative real values), the range of [tex]\( g(x) \)[/tex] is all real numbers except 0.
### Function [tex]\( h(x) = 3x \)[/tex]
1. Domain: The function [tex]\( h(x) = 3x \)[/tex] is a linear function, which means it is defined for all real numbers. Therefore, the domain of [tex]\( h(x) \)[/tex] is all real numbers.
2. Range: Since [tex]\( h(x) = 3x \)[/tex] is a linear function that can take any real value (as [tex]\( x \)[/tex] can be any real number), the range of [tex]\( h(x) \)[/tex] is all real numbers.
### Summary of Domain and Range
- Domain:
- [tex]\( f(x) \)[/tex]: all real numbers
- [tex]\( g(x) \)[/tex]: all real numbers except 0
- [tex]\( h(x) \)[/tex]: all real numbers
- Range:
- [tex]\( f(x) \)[/tex]: all non-negative real numbers
- [tex]\( g(x) \)[/tex]: all real numbers except 0
- [tex]\( h(x) \)[/tex]: all real numbers
### Conclusion
Based on the domain and range analysis, the accurate statements comparing the domain and range of the functions are:
1. The range of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] is all real numbers, but the range of [tex]\( g(x) \)[/tex] is all real numbers except 0. This statement is incorrect as [tex]\( f(x) \)[/tex] does not have a range of all real numbers but rather all non-negative real numbers.
2. The domain of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] is all real numbers, but the domain of [tex]\( g(x) \)[/tex] is all real numbers except 0. This statement is correct.
### Function [tex]\( f(x) = 3x^2 \)[/tex]
1. Domain: The function [tex]\( f(x) = 3x^2 \)[/tex] is a polynomial, which means it is defined for all real numbers. Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
2. Range: Since [tex]\( f(x) = 3x^2 \)[/tex] is a quadratic function that opens upwards (as the coefficient of [tex]\( x^2 \)[/tex] is positive), the minimum value of [tex]\( f(x) \)[/tex] is 0 (when [tex]\( x = 0 \)[/tex]). Thus, the range of [tex]\( f(x) \)[/tex] is all non-negative real numbers (i.e., [tex]\( [0, \infty) \)[/tex]).
### Function [tex]\( g(x) = \frac{1}{3x} \)[/tex]
1. Domain: The function [tex]\( g(x) = \frac{1}{3x} \)[/tex] involves division by [tex]\( x \)[/tex]. Division by zero is undefined, so [tex]\( x \)[/tex] cannot be 0. Therefore, the domain of [tex]\( g(x) \)[/tex] is all real numbers except 0.
2. Range: Since [tex]\( g(x) = \frac{1}{3x} \)[/tex] can never be 0 (it approaches infinity or negative infinity as [tex]\( x \)[/tex] gets close to 0, and it takes all positive and negative real values), the range of [tex]\( g(x) \)[/tex] is all real numbers except 0.
### Function [tex]\( h(x) = 3x \)[/tex]
1. Domain: The function [tex]\( h(x) = 3x \)[/tex] is a linear function, which means it is defined for all real numbers. Therefore, the domain of [tex]\( h(x) \)[/tex] is all real numbers.
2. Range: Since [tex]\( h(x) = 3x \)[/tex] is a linear function that can take any real value (as [tex]\( x \)[/tex] can be any real number), the range of [tex]\( h(x) \)[/tex] is all real numbers.
### Summary of Domain and Range
- Domain:
- [tex]\( f(x) \)[/tex]: all real numbers
- [tex]\( g(x) \)[/tex]: all real numbers except 0
- [tex]\( h(x) \)[/tex]: all real numbers
- Range:
- [tex]\( f(x) \)[/tex]: all non-negative real numbers
- [tex]\( g(x) \)[/tex]: all real numbers except 0
- [tex]\( h(x) \)[/tex]: all real numbers
### Conclusion
Based on the domain and range analysis, the accurate statements comparing the domain and range of the functions are:
1. The range of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] is all real numbers, but the range of [tex]\( g(x) \)[/tex] is all real numbers except 0. This statement is incorrect as [tex]\( f(x) \)[/tex] does not have a range of all real numbers but rather all non-negative real numbers.
2. The domain of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] is all real numbers, but the domain of [tex]\( g(x) \)[/tex] is all real numbers except 0. This statement is correct.
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