Question

1
select the correct answer.

the domain, range, and intercepts of a function are shown. which set of information could be characteristics of the functions inverse?
domain: $x \geq -1$\trange: $y \geq -2$
y-intercept: $(0,-1)$\tx-intercept: $(1,0)$

a. domain: $x \geq -2$; range: $y \geq -1$; y-intercept: $(0,1)$; x-intercept: $(-1,0)$
b. domain: $x \geq -1$; range: $y \geq -2$; y-intercept: $(0,-1)$; x-intercept: $(1,0)$
c. domain: $x \geq -1$; range: $y \geq -2$; y-intercept: $(1,0)$; x-intercept: $(0,-1)$
d. domain: $x \geq 1$; range: $y \geq 2$; y-intercept: $(0,1)$; x-intercept: $(1,0)$

Explanation:

Step1: Recall inverse function rules

For a function \(f(x)\) and its inverse \(f^{-1}(x)\):

• The domain of \(f^{-1}(x)\) = range of \(f(x)\)
• The range of \(f^{-1}(x)\) = domain of \(f(x)\)
• If \((a,b)\) is an intercept of \(f(x)\), then \((b,a)\) is an intercept of \(f^{-1}(x)\)

Step2: Apply rules to given function

Given original function:

• Domain: \(x \geq -1\), Range: \(y \geq -2\)
• y-intercept \((0,-1)\), x-intercept \((1,0)\)

For inverse function:

• Domain: \(x \geq -2\) (original range)
• Range: \(y \geq -1\) (original domain)
• y-intercept: \((-1,0)\) → swap to \((0,-1)\) becomes \((0,-1)\) swapped is \((-1,0)\)? No, correct swap: y-intercept of \(f(x)\) is \((0,-1)\), so x-intercept of \(f^{-1}(x)\) is \((-1,0)\); x-intercept of \(f(x)\) is \((1,0)\), so y-intercept of \(f^{-1}(x)\) is \((0,1)\)

Step3: Match to options

Check each option against derived inverse characteristics:

• Domain \(x \geq -2\), Range \(y \geq -1\), y-intercept \((0,1)\), x-intercept \((-1,0)\)

Answer:

A. domain: x ≥ -2; range: y ≥ -1; y-intercept (0,1); x-intercept (-1,0)