IDNLearn.com provides a seamless experience for finding accurate answers. Discover reliable and timely information on any topic from our network of experienced professionals.
Sagot :
To analyze the relationship between the functions [tex]\( f(x) = \sqrt{16} x \)[/tex] and [tex]\( g(x) = \sqrt[3]{64} \)[/tex], let's start by understanding each function separately.
1. Function [tex]\( f(x) = \sqrt{16} x \)[/tex]:
- Simplify the expression [tex]\(\sqrt{16}\)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
- Therefore, the function can be written as:
[tex]\[ f(x) = 4x \][/tex]
- This is a linear function with a slope of 4 and a y-intercept of 0. Its graph is a straight line passing through the origin (0,0) with a slope that indicates it rises 4 units for every 1 unit it moves to the right.
2. Function [tex]\( g(x) = \sqrt[3]{64} \)[/tex]:
- Simplify the expression [tex]\(\sqrt[3]{64}\)[/tex]:
[tex]\[ \sqrt[3]{64} = 4 \][/tex]
- Therefore, the function can be written as:
[tex]\[ g(x) = 4 \][/tex]
- This is a constant function. No matter the value of [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] always equals 4. Therefore, its graph is a horizontal line at [tex]\( y = 4 \)[/tex].
### Comparing the functions:
1. Initial Values:
- The initial value of [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex] is:
[tex]\[ f(0) = 4 \times 0 = 0 \][/tex]
- The initial value of [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex] is:
[tex]\[ g(0) = 4 \][/tex]
- Therefore, [tex]\( g(x) \)[/tex] has a greater initial value compared to [tex]\( f(x) \)[/tex].
2. Rate of Change:
- [tex]\( f(x) \)[/tex] increases linearly with a slope of 4. This means it increases at a constant rate.
- [tex]\( g(x) \)[/tex] is a constant function, which means it has no rate of change; it remains the same for all values of [tex]\( x \)[/tex].
- Since [tex]\( g(x) \)[/tex] does not change, it cannot be said to increase or decrease at any rate.
3. Equivalence and Other Properties:
- The functions are not equivalent in general since [tex]\( f(x) = 4x \)[/tex] changes with [tex]\( x \)[/tex], whereas [tex]\( g(x) = 4 \)[/tex] remains constant.
### Conclusion:
The correct inference from the analysis is that the function [tex]\( g(x) \)[/tex] has a greater initial value.
Therefore, the correct choice is:
- The function [tex]\( g(x) \)[/tex] has a greater initial value.
1. Function [tex]\( f(x) = \sqrt{16} x \)[/tex]:
- Simplify the expression [tex]\(\sqrt{16}\)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
- Therefore, the function can be written as:
[tex]\[ f(x) = 4x \][/tex]
- This is a linear function with a slope of 4 and a y-intercept of 0. Its graph is a straight line passing through the origin (0,0) with a slope that indicates it rises 4 units for every 1 unit it moves to the right.
2. Function [tex]\( g(x) = \sqrt[3]{64} \)[/tex]:
- Simplify the expression [tex]\(\sqrt[3]{64}\)[/tex]:
[tex]\[ \sqrt[3]{64} = 4 \][/tex]
- Therefore, the function can be written as:
[tex]\[ g(x) = 4 \][/tex]
- This is a constant function. No matter the value of [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] always equals 4. Therefore, its graph is a horizontal line at [tex]\( y = 4 \)[/tex].
### Comparing the functions:
1. Initial Values:
- The initial value of [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex] is:
[tex]\[ f(0) = 4 \times 0 = 0 \][/tex]
- The initial value of [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex] is:
[tex]\[ g(0) = 4 \][/tex]
- Therefore, [tex]\( g(x) \)[/tex] has a greater initial value compared to [tex]\( f(x) \)[/tex].
2. Rate of Change:
- [tex]\( f(x) \)[/tex] increases linearly with a slope of 4. This means it increases at a constant rate.
- [tex]\( g(x) \)[/tex] is a constant function, which means it has no rate of change; it remains the same for all values of [tex]\( x \)[/tex].
- Since [tex]\( g(x) \)[/tex] does not change, it cannot be said to increase or decrease at any rate.
3. Equivalence and Other Properties:
- The functions are not equivalent in general since [tex]\( f(x) = 4x \)[/tex] changes with [tex]\( x \)[/tex], whereas [tex]\( g(x) = 4 \)[/tex] remains constant.
### Conclusion:
The correct inference from the analysis is that the function [tex]\( g(x) \)[/tex] has a greater initial value.
Therefore, the correct choice is:
- The function [tex]\( g(x) \)[/tex] has a greater initial value.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
