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Let's analyze the graph of the function [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] to determine which statements are correct.
1. The graph has a domain of all real numbers.
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. This means that the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex] is also defined for all real numbers because we are simply translating the cube root function horizontally. Adding 2 does not affect the domain. Therefore, the domain of [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] is all real numbers. This statement is true.
2. The graph has a range of [tex]\( y \geq 1 \)[/tex].
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] can produce any real number. After translating horizontally by 1 and adding 2, the range remains all real numbers because [tex]\( \sqrt[3]{x - 1} \)[/tex] can still yield any real number and adding 2 shifts the range but does not restrict it. Therefore, the range is all real numbers, not [tex]\( y \geq 1 \)[/tex]. This statement is false.
3. As [tex]\( x \)[/tex] is increasing, [tex]\( y \)[/tex] is decreasing.
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is an increasing function. When [tex]\( x \)[/tex] increases, [tex]\( \sqrt[3]{x - 1} \)[/tex] will also increase, which in turn means [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] will increase. Therefore, as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases. This statement is false.
4. The graph has a [tex]\( y \)[/tex]-intercept at [tex]\( (0, 1) \)[/tex].
To find the [tex]\( y \)[/tex]-intercept, we set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt[3]{0 - 1} + 2 = \sqrt[3]{-1} + 2 = -1 + 2 = 1 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is at [tex]\( (0, 1) \)[/tex]. This statement is true.
5. The graph has an [tex]\( x \)[/tex]-intercept at [tex]\( (-7, 0) \)[/tex].
To find the [tex]\( x \)[/tex]-intercept, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = \sqrt[3]{x - 1} + 2 \implies \sqrt[3]{x - 1} = -2 \][/tex]
Cubing both sides, we get:
[tex]\[ x - 1 = (-2)^3 = -8 \implies x = -8 + 1 = -7 \][/tex]
Therefore, the [tex]\( x \)[/tex]-intercept is at [tex]\( (-7, 0) \)[/tex]. This statement is true.
Based on this analysis, the three correct statements are:
1. The graph has a domain of all real numbers.
2. The graph has a [tex]\( y \)[/tex]-intercept at [tex]\( (0, 1) \)[/tex].
3. The graph has an [tex]\( x \)[/tex]-intercept at [tex]\( (-7, 0) \)[/tex].
1. The graph has a domain of all real numbers.
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. This means that the function [tex]\( y = \sqrt[3]{x - 1} \)[/tex] is also defined for all real numbers because we are simply translating the cube root function horizontally. Adding 2 does not affect the domain. Therefore, the domain of [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] is all real numbers. This statement is true.
2. The graph has a range of [tex]\( y \geq 1 \)[/tex].
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] can produce any real number. After translating horizontally by 1 and adding 2, the range remains all real numbers because [tex]\( \sqrt[3]{x - 1} \)[/tex] can still yield any real number and adding 2 shifts the range but does not restrict it. Therefore, the range is all real numbers, not [tex]\( y \geq 1 \)[/tex]. This statement is false.
3. As [tex]\( x \)[/tex] is increasing, [tex]\( y \)[/tex] is decreasing.
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is an increasing function. When [tex]\( x \)[/tex] increases, [tex]\( \sqrt[3]{x - 1} \)[/tex] will also increase, which in turn means [tex]\( y = \sqrt[3]{x - 1} + 2 \)[/tex] will increase. Therefore, as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases. This statement is false.
4. The graph has a [tex]\( y \)[/tex]-intercept at [tex]\( (0, 1) \)[/tex].
To find the [tex]\( y \)[/tex]-intercept, we set [tex]\( x = 0 \)[/tex] and solve for [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt[3]{0 - 1} + 2 = \sqrt[3]{-1} + 2 = -1 + 2 = 1 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is at [tex]\( (0, 1) \)[/tex]. This statement is true.
5. The graph has an [tex]\( x \)[/tex]-intercept at [tex]\( (-7, 0) \)[/tex].
To find the [tex]\( x \)[/tex]-intercept, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = \sqrt[3]{x - 1} + 2 \implies \sqrt[3]{x - 1} = -2 \][/tex]
Cubing both sides, we get:
[tex]\[ x - 1 = (-2)^3 = -8 \implies x = -8 + 1 = -7 \][/tex]
Therefore, the [tex]\( x \)[/tex]-intercept is at [tex]\( (-7, 0) \)[/tex]. This statement is true.
Based on this analysis, the three correct statements are:
1. The graph has a domain of all real numbers.
2. The graph has a [tex]\( y \)[/tex]-intercept at [tex]\( (0, 1) \)[/tex].
3. The graph has an [tex]\( x \)[/tex]-intercept at [tex]\( (-7, 0) \)[/tex].
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