Question

The function f is defined by the rule that assigns each real number x to 1 more than the opposite of its cube root. Select all the true statements about the function f.

A. The domain is all real numbers.
B. The range is all real numbers.
C. The x-intercept is 0.
D. The y-intercept is 1.
E. The graph is decreasing for all values in the domain of f.

Answer 1

• A, B, D, E

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As per given definition the function is:

• 
  `f(x) = -\sqrt[3]{x} + 1`

Verify each statement:

A. The domain is all real numbers ⇒ TRUE, as there is no restrictions to the domain.

B. The range is all real numbers ⇒ TRUE, since no domain restrictions, the range has no restrictions.

C. The x-intercept is 0 ⇒ FALSE

It is the point with zero y-coordinate:

• 
  `0= -\sqrt[3]{x} + 1`
• 
  `\sqrt[3]{x} = 1`
• 
  `x=1`

D. The y-intercept is 1 ⇒ TRUE

It is the point with zero x-coordinate:

• 
  `f(0)=-\sqrt[3]{0}+1=0+1=1`

E. The graph is decreasing for all values in the domain of f ⇒ TRUE

The domain has a negative coefficient and the function is cube - root function, hence the odd function, therefore it is always decreasing.

See attached graph to help with answer choices.