Answer :
Let's analyze the given function [tex]\( y = \sqrt[3]{x-1} + 2 \)[/tex] step by step to determine which statements about its graph are true.
1. The graph has a domain of all real numbers.
The cube root function [tex]\( \sqrt[3]{x-1} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. This means that no matter what value you choose for [tex]\( x \)[/tex], you can always find its cube root. Therefore, the entire expression [tex]\( \sqrt[3]{x-1} + 2 \)[/tex] is defined for all real numbers.
Conclusion: The graph has a domain of all real numbers.
2. The graph has a range of [tex]\( y \geq 1 \)[/tex].
We need to find the range of [tex]\( y = \sqrt[3]{x-1} + 2 \)[/tex]. The cube root function [tex]\( \sqrt[3]{x-1} \)[/tex] can take on any real value (from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex]). Adding 2 shifts the entire range up by 2 but does not place any restriction on the range.
Conclusion: The graph does not have a range of [tex]\( y \geq 1 \)[/tex]; instead, it has the range of all real numbers.
3. As [tex]\( x \)[/tex] is increasing, [tex]\( y \)[/tex] is decreasing.
To determine how [tex]\( y \)[/tex] changes as [tex]\( x \)[/tex] increases, consider the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{dy}{dx} = \frac{d}{dx} (\sqrt[3]{x-1} + 2) = \frac{1}{3} (x-1)^{-\frac{2}{3}} \][/tex]
The derivative [tex]\( \frac{1}{3} (x-1)^{-\frac{2}{3}} \)[/tex] is positive for all [tex]\( x \neq 1 \)[/tex]. Hence, [tex]\( y \)[/tex] is increasing as [tex]\( x \)[/tex] increases.
Conclusion: As [tex]\( x \)[/tex] is increasing, [tex]\( y \)[/tex] is also increasing, not decreasing.
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]
Thus, the graph intersects the [tex]\( y \)[/tex]-axis at [tex]\( (0, 1) \)[/tex].
Conclusion: The graph has a [tex]\( y \)[/tex]-intercept at [tex]\( (0,1) \)[/tex].
5. The graph has an [tex]\( x \)[/tex]-intercept at [tex]\( (-7,0) \)[/tex].
To find the [tex]\( x \)[/tex]-intercept, set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = \sqrt[3]{x-1} + 2 \implies \sqrt[3]{x-1} = -2 \implies x - 1 = -8 \implies x = -7 \][/tex]
Thus, the graph intersects the [tex]\( x \)[/tex]-axis at [tex]\( (-7, 0) \)[/tex].
Conclusion: The graph has an [tex]\( x \)[/tex]-intercept at [tex]\( (-7,0) \)[/tex].
Based on this analysis, the three statements that are true 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]\( \sqrt[3]{x-1} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. This means that no matter what value you choose for [tex]\( x \)[/tex], you can always find its cube root. Therefore, the entire expression [tex]\( \sqrt[3]{x-1} + 2 \)[/tex] is defined for all real numbers.
Conclusion: The graph has a domain of all real numbers.
2. The graph has a range of [tex]\( y \geq 1 \)[/tex].
We need to find the range of [tex]\( y = \sqrt[3]{x-1} + 2 \)[/tex]. The cube root function [tex]\( \sqrt[3]{x-1} \)[/tex] can take on any real value (from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex]). Adding 2 shifts the entire range up by 2 but does not place any restriction on the range.
Conclusion: The graph does not have a range of [tex]\( y \geq 1 \)[/tex]; instead, it has the range of all real numbers.
3. As [tex]\( x \)[/tex] is increasing, [tex]\( y \)[/tex] is decreasing.
To determine how [tex]\( y \)[/tex] changes as [tex]\( x \)[/tex] increases, consider the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{dy}{dx} = \frac{d}{dx} (\sqrt[3]{x-1} + 2) = \frac{1}{3} (x-1)^{-\frac{2}{3}} \][/tex]
The derivative [tex]\( \frac{1}{3} (x-1)^{-\frac{2}{3}} \)[/tex] is positive for all [tex]\( x \neq 1 \)[/tex]. Hence, [tex]\( y \)[/tex] is increasing as [tex]\( x \)[/tex] increases.
Conclusion: As [tex]\( x \)[/tex] is increasing, [tex]\( y \)[/tex] is also increasing, not decreasing.
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]
Thus, the graph intersects the [tex]\( y \)[/tex]-axis at [tex]\( (0, 1) \)[/tex].
Conclusion: The graph has a [tex]\( y \)[/tex]-intercept at [tex]\( (0,1) \)[/tex].
5. The graph has an [tex]\( x \)[/tex]-intercept at [tex]\( (-7,0) \)[/tex].
To find the [tex]\( x \)[/tex]-intercept, set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = \sqrt[3]{x-1} + 2 \implies \sqrt[3]{x-1} = -2 \implies x - 1 = -8 \implies x = -7 \][/tex]
Thus, the graph intersects the [tex]\( x \)[/tex]-axis at [tex]\( (-7, 0) \)[/tex].
Conclusion: The graph has an [tex]\( x \)[/tex]-intercept at [tex]\( (-7,0) \)[/tex].
Based on this analysis, the three statements that are true 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].
