Answer :
To determine which statements correctly describe the graph of [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex], let's analyze its properties:
1. Domain:
- The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers. Since our function involves a cube root shifted by 1 inside, it remains defined for all [tex]\( x \)[/tex].
- Therefore, the domain of [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] is all real numbers.
2. Range:
- The cube root function [tex]\( \sqrt[3]{x} \)[/tex] can take any real number value, as it is defined for all real numbers.
- Adding 2 shifts the entire range up by 2 units, but the overall set of possible [tex]\( y \)[/tex]-values remains all real numbers.
- Thus, the range of [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] is also all real numbers.
3. Behavior as [tex]\( x \)[/tex] increases:
- The function [tex]\( \sqrt[3]{x - 1} \)[/tex] is an increasing function, meaning as [tex]\( x \)[/tex] increases, [tex]\( \sqrt[3]{x - 1} \)[/tex] also increases.
- Adding 2 to [tex]\( \sqrt[3]{x - 1} \)[/tex] just shifts the graph vertically but does not affect its increasing nature.
- Thus, as [tex]\( x \)[/tex] increases, [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] also increases.
4. [tex]\( y \)[/tex]-intercept:
- The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the equation [tex]\( y = \sqrt[3]{0 - 1} + 2 \)[/tex]:
[tex]\[ y = \sqrt[3]{-1} + 2 = -1 + 2 = 1 \][/tex]
- Therefore, the [tex]\( y \)[/tex]-intercept is at [tex]\( (0, 1) \)[/tex].
5. [tex]\( x \)[/tex]-intercept:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( y = 0 \)[/tex].
- Setting [tex]\( y = 0 \)[/tex] in the equation [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex]:
[tex]\[ 0 = \sqrt[3]{x - 1} + 2 \implies \sqrt[3]{x - 1} = -2 \][/tex]
[tex]\[ x - 1 = -8 \implies x = -7 \][/tex]
- Therefore, the [tex]\( x \)[/tex]-intercept is at [tex]\( (-7, 0) \)[/tex].
Given these analyses, the three correct statements are:
- The graph has a domain of all real numbers.
- The graph has a [tex]\( y \)[/tex]-intercept at [tex]\( (0, 1) \)[/tex].
- The graph has an [tex]\( x \)[/tex]-intercept at [tex]\( (-7, 0) \)[/tex].
1. Domain:
- The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers. Since our function involves a cube root shifted by 1 inside, it remains defined for all [tex]\( x \)[/tex].
- Therefore, the domain of [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] is all real numbers.
2. Range:
- The cube root function [tex]\( \sqrt[3]{x} \)[/tex] can take any real number value, as it is defined for all real numbers.
- Adding 2 shifts the entire range up by 2 units, but the overall set of possible [tex]\( y \)[/tex]-values remains all real numbers.
- Thus, the range of [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] is also all real numbers.
3. Behavior as [tex]\( x \)[/tex] increases:
- The function [tex]\( \sqrt[3]{x - 1} \)[/tex] is an increasing function, meaning as [tex]\( x \)[/tex] increases, [tex]\( \sqrt[3]{x - 1} \)[/tex] also increases.
- Adding 2 to [tex]\( \sqrt[3]{x - 1} \)[/tex] just shifts the graph vertically but does not affect its increasing nature.
- Thus, as [tex]\( x \)[/tex] increases, [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] also increases.
4. [tex]\( y \)[/tex]-intercept:
- The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the equation [tex]\( y = \sqrt[3]{0 - 1} + 2 \)[/tex]:
[tex]\[ y = \sqrt[3]{-1} + 2 = -1 + 2 = 1 \][/tex]
- Therefore, the [tex]\( y \)[/tex]-intercept is at [tex]\( (0, 1) \)[/tex].
5. [tex]\( x \)[/tex]-intercept:
- The [tex]\( x \)[/tex]-intercept occurs where [tex]\( y = 0 \)[/tex].
- Setting [tex]\( y = 0 \)[/tex] in the equation [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex]:
[tex]\[ 0 = \sqrt[3]{x - 1} + 2 \implies \sqrt[3]{x - 1} = -2 \][/tex]
[tex]\[ x - 1 = -8 \implies x = -7 \][/tex]
- Therefore, the [tex]\( x \)[/tex]-intercept is at [tex]\( (-7, 0) \)[/tex].
Given these analyses, the three correct statements are:
- The graph has a domain of all real numbers.
- The graph has a [tex]\( y \)[/tex]-intercept at [tex]\( (0, 1) \)[/tex].
- The graph has an [tex]\( x \)[/tex]-intercept at [tex]\( (-7, 0) \)[/tex].
