Question

guided practice
inverse relations and functions
which of the following describes the domain and range of the function ( m(x) = 12 - sqrt{x} ) and its inverse?
a.

	function	inverse
domain	( x geq 0 )	( x geq 0 )
range	( y leq 12 )	( y leq 12 )

b.

	function	inverse
domain	( x geq 0 )	( x leq 12 )
range	( y leq 12 )	( y geq 0 )

c.

	function	inverse
domain	( x leq 12 )	( x leq 12 )
range	( y geq 0 )	( y geq 0 )

d.

	function	inverse
domain	all real numbers	( x leq 12 )
range	all real numbers	( y geq 0 )

Explanation:

Step1: Analyze the original function \( y = 12-\sqrt{x} \)

• Domain of the function: The square root function \( \sqrt{x} \) is defined when \( x\geq0 \), so the domain of \( y = 12-\sqrt{x} \) is \( x\geq0 \).
• Range of the function: Since \( \sqrt{x}\geq0 \), then \( -\sqrt{x}\leq0 \). Adding 12 to both sides, we get \( y = 12-\sqrt{x}\leq12 \). So the range of the function is \( y\leq12 \).

Step2: Find the inverse function and analyze its domain and range

• First, swap \( x \) and \( y \) in \( y = 12-\sqrt{x} \): \( x = 12-\sqrt{y} \).
• Then solve for \( y \):
• Rearrange the equation: \( \sqrt{y}=12 - x \).
• Square both sides: \( y=(12 - x)^2 \).
• Domain of the inverse function: The domain of the inverse function is the range of the original function. Since the range of the original function is \( y\leq12 \), the domain of the inverse function is \( x\leq12 \).
• Range of the inverse function: The range of the inverse function is the domain of the original function. Since the domain of the original function is \( x\geq0 \), the range of the inverse function is \( y\geq0 \).
• Comparing with the options, option B has the function's domain as \( x\geq0 \), function's range as \( y\leq12 \), inverse's domain as \( x\leq12 \) and inverse's range as \( y\geq0 \) (matching our analysis, considering possible table - formatting inconsistencies in the option's display).

Answer:

B. Function: Domain \( x \geq 0 \), Range \( y \leq 12 \); Inverse: Domain \( x \leq 12 \), Range \( y \geq 0 \) (Note: The original option B's table likely has a typo in the inverse domain/range labeling, but based on the function \( y = 12 - \sqrt{x} \), the correct domain of the function is \( x \geq 0 \) (since square root requires non - negative input), the range of the function: as \( \sqrt{x}\geq0 \), then \( - \sqrt{x}\leq0 \), so \( y = 12-\sqrt{x}\leq12 \). To find the inverse, we first swap \( x \) and \( y \): \( x = 12-\sqrt{y} \), then solve for \( y \): \( \sqrt{y}=12 - x \), so \( y=(12 - x)^2 \). For the inverse function, the domain of the inverse is the range of the original function (\( x\leq12 \)) and the range of the inverse is the domain of the original function (\( y\geq0 \)), which matches option B's structure when considering possible table formatting issues.)