Question

a function f(x) = x² - 2. which of the following describes the domain and range of the function and its inverse?
(three tables with options for function and inverse domain and range are shown:
first table: function domain: all real numbers, function range: all real numbers; inverse domain: all real numbers, inverse range: all real numbers.
second table: function domain: all real numbers, function range: y ≥ -2; inverse domain: x ≥ -2, inverse range: all real numbers.
third table: function domain: x ≥ 0, function range: y ≥ -2; inverse domain: x ≥ -2, inverse range: y ≥ 0.)

Explanation:

Step1: Analyze the function \( f(x)=x^2 - 2 \)

For the function \( y = x^2-2 \), the domain of a quadratic function \( ax^2+bx + c \) (here \( a = 1,b = 0,c=-2\)) is all real numbers? Wait, no, wait. Wait, if we consider the function \( f(x)=x^2 - 2 \), actually, the domain of \( y=x^2-2 \) is all real numbers, but the range: since \( x^2\geq0 \), then \( y=x^2 - 2\geq - 2 \). But wait, maybe the function is a parabola, but if we are talking about the inverse, we need to consider the function's restrict domain. Wait, maybe the original function is \( f(x)=x^2 - 2 \) with domain \( x\geq0 \)? Wait, no, let's check the tables.

Wait the right table: Function domain \( x\geq0 \), range \( y\geq - 2 \); Inverse domain \( x\geq - 2 \), range \( y\geq0 \). Let's find the inverse of \( y=x^2 - 2 \) (assuming \( x\geq0 \)). Solve for \( x \): \( y + 2=x^2\), so \( x=\sqrt{y + 2} \) (since \( x\geq0 \)), so the inverse function is \( f^{-1}(x)=\sqrt{x + 2} \). For the inverse function \( f^{-1}(x)=\sqrt{x + 2} \), the domain of the inverse is \( x+2\geq0\Rightarrow x\geq - 2 \), and the range is \( y\geq0 \) (since square root is non - negative). And for the original function \( f(x)=x^2 - 2 \) with \( x\geq0 \), the domain is \( x\geq0 \), and the range is \( y=x^2 - 2\geq - 2 \) (since \( x\geq0,x^2\geq0,y\geq - 2 \)). So the right table matches: function domain \( x\geq0 \), range \( y\geq - 2 \); inverse domain \( x\geq - 2 \), range \( y\geq0 \).

Answer:

The Right Table