True statements:
The domain of h(x) is the set of all real numbers.
The range of h(x) is the set of all real numbers
Step-by-step explanation:
The function given in this problem is
[tex]h(x)=\sqrt[3]{x-4}[/tex]
Now let's analyze the statements given:
- The domain of h(x) is the set of all real numbers. --> TRUE. The domain of a function is the set of values allowed for x. Here there is no restriction for the value of x, since the argument of an odd root can have any value (either positive or negative), so the domain of h(x) is the set of all real numbers.
- The range of h(x) is the set of all real numbers. --> TRUE. The range of a function is the set of values that the y can take. In this case, y can take any value: in fact, as [tex]x \rightarrow +\infty[/tex], then [tex]y \rightarrow +\infty[/tex], and as [tex]x \rightarrow -\infty[/tex], then [tex]y \rightarrow -\infty[/tex], and since there are no points of discontinuity in the function, this means that it can take any value of y.
For all points (x, h(x)), h(x) exists if and only if x – 4 > 0 --> FALSE. As we stated in part 1), the argument of a cubic root can be either positive or negative, so the function is defined also for x - 4 < 0.
The graph of h(x) is a translation of f(x) down 4 units. --> FALSE. The form of f(x) is not given so we can't evaluate this statement.
The graph of h(x) intercepts the x-axis at (4, 0). --> FALSE. In fact, when [tex]x=0[/tex], the value of the function is
[tex]h(0)=\sqrt[3]{0-4}=\sqrt[3]{-4}[/tex]
which is different from 4.
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Answer:
A. The domain of h(x) is the set of all real numbers.
B. The range of h(x) is the set of all real numbers.
E. The graph of h(x) intercepts the x-axis at (4, 0).
