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Below is a detailed, step-by-step solution for comparing the domain and range of the functions [tex]\( f(x) = 3 x^2 \)[/tex], [tex]\( g(x) = \frac{1}{3x} \)[/tex], and [tex]\( h(x) = 3 x \)[/tex].
### Step 1: Analyzing the Domain of the Functions
1. Domain of [tex]\( f(x) = 3 x^2 \)[/tex]:
- The function [tex]\( f(x) = 3 x^2 \)[/tex] is a polynomial, and polynomials are defined for all real numbers.
- Thus, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
2. Domain of [tex]\( g(x) = \frac{1}{3x} \)[/tex]:
- The function [tex]\( g(x) = \frac{1}{3x} \)[/tex] is a rational function.
- Rational functions are defined for all real numbers except where the denominator is zero.
- For [tex]\( g(x) = \frac{1}{3x} \)[/tex], the denominator is zero when [tex]\( x = 0 \)[/tex].
- Thus, the domain of [tex]\( g(x) \)[/tex] is all real numbers except 0.
3. Domain of [tex]\( h(x) = 3 x \)[/tex]:
- The function [tex]\( h(x) = 3 x \)[/tex] is a linear function.
- Linear functions are defined for all real numbers.
- Thus, the domain of [tex]\( h(x) \)[/tex] is all real numbers.
### Step 2: Analyzing the Range of the Functions
1. Range of [tex]\( f(x) = 3 x^2 \)[/tex]:
- The function [tex]\( f(x) = 3 x^2 \)[/tex] is a parabola that opens upwards.
- A parabola of the form [tex]\( ax^2 \)[/tex] has its minimum value at the vertex.
- For [tex]\( 3 x^2 \)[/tex], the minimum value is 0 (when [tex]\( x = 0 \)[/tex]).
- Since the parabola opens upwards, the range is all non-negative real numbers ([tex]\( y \geq 0 \)[/tex]).
- Thus, the range of [tex]\( f(x) \)[/tex] is all non-negative real numbers.
2. Range of [tex]\( g(x) = \frac{1}{3x} \)[/tex]:
- The function [tex]\( g(x) = \frac{1}{3x} \)[/tex] is a hyperbola.
- As [tex]\( x \)[/tex] approaches 0 (either from the positive or negative side), [tex]\( \frac{1}{3x} \)[/tex] approaches ±∞.
- As [tex]\( x \)[/tex] moves away from 0, [tex]\( \frac{1}{3x} \)[/tex] covers all real values except 0.
- Thus, the range of [tex]\( g(x) \)[/tex] is all real numbers except 0.
3. Range of [tex]\( h(x) = 3 x \)[/tex]:
- The function [tex]\( h(x) = 3 x \)[/tex] is a linear function.
- Linear functions cover all real values along the y-axis as [tex]\( x \)[/tex] varies.
- Thus, the range of [tex]\( h(x) \)[/tex] is all real numbers.
### Step 3: Comparing Statements
Let's compare the given statements based on our analysis:
1. All of the functions have a unique range.
- This statement is not accurate because the ranges of [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different.
2. The range of all three functions is all real numbers.
- This statement is not accurate because the range of [tex]\( f(x) \)[/tex] is all non-negative real numbers, not all real numbers.
3. The domain of all three functions is all real numbers.
- This statement is not accurate because the domain of [tex]\( g(x) \)[/tex] is all real numbers except 0.
4. 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 not accurate because the range of [tex]\( f(x) \)[/tex] is all non-negative real numbers, not all real numbers.
5. 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 accurate based on our analysis.
### Conclusion
The two accurate statements that compare the domain and range of the given functions are:
- 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.
### Step 1: Analyzing the Domain of the Functions
1. Domain of [tex]\( f(x) = 3 x^2 \)[/tex]:
- The function [tex]\( f(x) = 3 x^2 \)[/tex] is a polynomial, and polynomials are defined for all real numbers.
- Thus, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
2. Domain of [tex]\( g(x) = \frac{1}{3x} \)[/tex]:
- The function [tex]\( g(x) = \frac{1}{3x} \)[/tex] is a rational function.
- Rational functions are defined for all real numbers except where the denominator is zero.
- For [tex]\( g(x) = \frac{1}{3x} \)[/tex], the denominator is zero when [tex]\( x = 0 \)[/tex].
- Thus, the domain of [tex]\( g(x) \)[/tex] is all real numbers except 0.
3. Domain of [tex]\( h(x) = 3 x \)[/tex]:
- The function [tex]\( h(x) = 3 x \)[/tex] is a linear function.
- Linear functions are defined for all real numbers.
- Thus, the domain of [tex]\( h(x) \)[/tex] is all real numbers.
### Step 2: Analyzing the Range of the Functions
1. Range of [tex]\( f(x) = 3 x^2 \)[/tex]:
- The function [tex]\( f(x) = 3 x^2 \)[/tex] is a parabola that opens upwards.
- A parabola of the form [tex]\( ax^2 \)[/tex] has its minimum value at the vertex.
- For [tex]\( 3 x^2 \)[/tex], the minimum value is 0 (when [tex]\( x = 0 \)[/tex]).
- Since the parabola opens upwards, the range is all non-negative real numbers ([tex]\( y \geq 0 \)[/tex]).
- Thus, the range of [tex]\( f(x) \)[/tex] is all non-negative real numbers.
2. Range of [tex]\( g(x) = \frac{1}{3x} \)[/tex]:
- The function [tex]\( g(x) = \frac{1}{3x} \)[/tex] is a hyperbola.
- As [tex]\( x \)[/tex] approaches 0 (either from the positive or negative side), [tex]\( \frac{1}{3x} \)[/tex] approaches ±∞.
- As [tex]\( x \)[/tex] moves away from 0, [tex]\( \frac{1}{3x} \)[/tex] covers all real values except 0.
- Thus, the range of [tex]\( g(x) \)[/tex] is all real numbers except 0.
3. Range of [tex]\( h(x) = 3 x \)[/tex]:
- The function [tex]\( h(x) = 3 x \)[/tex] is a linear function.
- Linear functions cover all real values along the y-axis as [tex]\( x \)[/tex] varies.
- Thus, the range of [tex]\( h(x) \)[/tex] is all real numbers.
### Step 3: Comparing Statements
Let's compare the given statements based on our analysis:
1. All of the functions have a unique range.
- This statement is not accurate because the ranges of [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different.
2. The range of all three functions is all real numbers.
- This statement is not accurate because the range of [tex]\( f(x) \)[/tex] is all non-negative real numbers, not all real numbers.
3. The domain of all three functions is all real numbers.
- This statement is not accurate because the domain of [tex]\( g(x) \)[/tex] is all real numbers except 0.
4. 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 not accurate because the range of [tex]\( f(x) \)[/tex] is all non-negative real numbers, not all real numbers.
5. 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 accurate based on our analysis.
### Conclusion
The two accurate statements that compare the domain and range of the given functions are:
- 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.
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