Question

select the correct answer.
the domain, range, and intercepts of a function are shown. which set of information could be characteristics of the function’s inverse?
domain: ( x geq -2 ) range: ( y geq -1 )
y-intercept: ( (0,4) ) x-intercept: ( (8,0) )

a. domain: ( x geq -1 ), range: ( y geq -2 ), y-intercept: ( (0,8) ), x-intercept: ( (4,0) )
b. domain: ( x geq -1 ), range: ( y geq -2 ), y-intercept: ( (4,0) ), x-intercept: ( (0,8) )
c. domain: ( x geq -2 ), range: ( y geq -1 ), y-intercept: ( (0,8) ), x-intercept: ( (4,0) )
d. domain: ( x geq -2 ), range: ( y geq -1 ), y-intercept: ( (0,4) ), x-intercept: ( (8,0) )

Explanation:

Step1: Recall inverse function properties

For a function \( f(x) \) and its inverse \( f^{-1}(x) \), the domain of \( f \) is the range of \( f^{-1} \), and the range of \( f \) is the domain of \( f^{-1} \). Also, if \( (a,b) \) is on \( f(x) \), then \( (b,a) \) is on \( f^{-1}(x) \).

Step2: Analyze domain and range of inverse

Given the original function has domain \( x \geq -2 \) and range \( y \geq -1 \). So the inverse function should have domain \( x \geq -1 \) (range of original) and range \( y \geq -2 \) (domain of original).

Step3: Analyze intercepts of inverse

Original \( y \)-intercept: \( (0,4) \), so inverse should have \( x \)-intercept \( (4,0) \) (since \( (0,4) \) on \( f \) implies \( (4,0) \) on \( f^{-1} \)). Original \( x \)-intercept: \( (8,0) \), so inverse should have \( y \)-intercept \( (0,8) \) (since \( (8,0) \) on \( f \) implies \( (0,8) \) on \( f^{-1} \)).

Step4: Match with options

Check each option:

• Option A: Domain \( x \geq -1 \), range \( y \geq -2 \), \( y \)-intercept \( (0,8) \), \( x \)-intercept \( (4,0) \). This matches the properties.
• Option B: \( y \)-intercept \( (4,0) \) and \( x \)-intercept \( (0,8) \) are swapped, incorrect.
• Option C: Domain and range not swapped, incorrect.
• Option D: Domain and range not swapped, intercepts same as original, incorrect.

Answer:

A. domain: \( x \geq -1 \), range: \( y \geq -2 \), \( y \)-intercept: \( (0,8) \), \( x \)-intercept: \( (4,0) \)