Consider the functions [tex]f(x)=3x^2[/tex], [tex]g(x)=\frac{1}{3x}[/tex], and [tex]h(x)=3x[/tex].

Which statements accurately compare the domain and range of the functions? Select two options.

A. All of the functions have a unique range.
B. The range of all three functions is all real numbers.
C. The domain of all three functions is all real numbers.
D. 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.
E. 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.



Answer :

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.