Let's analyze each statement given about the graph of the function [tex]\( h(x) = \sqrt[3]{x-4} \)[/tex] to determine which statements are true.
1. The domain of [tex]\( h(x) \)[/tex] is the set of all real numbers.
The function [tex]\( h(x) = \sqrt[3]{x-4} \)[/tex] involves the cube root of [tex]\( x-4 \)[/tex]. The cube root function is defined for all real numbers. Therefore, the domain of [tex]\( h(x) \)[/tex] includes all real numbers.
True.
2. The range of [tex]\( h(x) \)[/tex] is the set of all real numbers.
The cube root function, [tex]\( \sqrt[3]{x} \)[/tex], can output any real number based on the input [tex]\( x \)[/tex]. Since [tex]\( h(x) = \sqrt[3]{x-4} \)[/tex] only involves shifting the input [tex]\( x \)[/tex] by 4, it does not change the fact that [tex]\( h(x) \)[/tex] can output any real number.
True.
3. For all points [tex]\( (x, h(x)) \)[/tex], [tex]\( h(x) \)[/tex] exists if and only if [tex]\( x-4 \geq 0 \)[/tex].
The cube root function [tex]\( \sqrt[3]{x} \)[/tex] is defined for all real numbers, whether [tex]\( x \)[/tex] is positive, negative, or zero. Thus, [tex]\( h(x) = \sqrt[3]{x-4} \)[/tex] is also defined for all values of [tex]\( x \)[/tex], not just those where [tex]\( x-4 \geq 0 \)[/tex].
False.
4. The graph of [tex]\( h(x) \)[/tex] is a translation of [tex]\( f(x) \downarrow 4 \)[/tex] units.
The given function [tex]\( h(x) = \sqrt[3]{x-4} \)[/tex] represents a horizontal transformation, where the graph of [tex]\( y = \sqrt[3]{x} \)[/tex] is shifted to the right by 4 units. Therefore, it is not a translation downwards.
False.
5. The graph of [tex]\( h(x) \)[/tex] intercepts the [tex]\( x \)[/tex]-axis at [tex]\( (4,0) \)[/tex].
To find the x-intercept, set [tex]\( h(x) = 0 \)[/tex]:
[tex]\[
0 = \sqrt[3]{x-4}
\][/tex]
This implies [tex]\( x-4 = 0 \)[/tex], so [tex]\( x = 4 \)[/tex]. Thus, the graph intercepts the x-axis at [tex]\( (4,0) \)[/tex].
True.
The summary of which statements are true:
- The domain of [tex]\( h(x) \)[/tex] is the set of all real numbers.
- The range of [tex]\( h(x) \)[/tex] is the set of all real numbers.
- The graph of [tex]\( h(x) \)[/tex] intercepts the [tex]\( x \)[/tex]-axis at [tex]\( (4,0) \)[/tex].
And which are false:
- For all points [tex]\( (x, h(x)) \)[/tex], [tex]\( h(x) \)[/tex] exists if and only if [tex]\( x-4 \geq 0 \)[/tex].
- The graph of [tex]\( h(x) \)[/tex] is a translation of [tex]\( f(x) \downarrow 4 \)[/tex] units.
