Answer:
f(x)
Domain: all real numbers
Range:[tex]y\ge -2[/tex]
[tex]f^{-1}(x)[/tex]
Domain: [tex]y\ge -2[/tex]
Range:all real numbers
Step-by-step explanation:
Given [tex]f(x)=x^2-2[/tex]
The domain of this function, refers to all values of x for which f(x) is defined.
The given function is a quadratic polynomial.
Polynomial functions are defined everywhere, therefore the domain is all real numbers.
The range refers to the values of y, for which x is defined
Let [tex]y=x^2-2[/tex]
Solve for x;
[tex]x=\pm \sqrt{y+2}[/tex]
This function is defined for [tex]y+2\ge0[/tex]
This implies that; [tex]y\ge-2[/tex]
Inverse Function
The domain of f(x) becomes the range of [tex]f^{-1}(x)[/tex] and the range of f(x) becomes the domain of [tex]f^{-1}(x)[/tex].
Domain: [tex]y\ge -2[/tex]
Range:all real numbers
Or see it in details
Let [tex]y=x^2-2[/tex]
interchange x and y;
[tex]x=y^2-2[/tex]
[tex]x+2=y^2[/tex]
[tex]\pm \sqrt{x+2}=y[/tex]
[tex]f^{-1}(x)=\pm \sqrt{x+2}[/tex]
Domain: [tex]x\ge-2[/tex]
Range:
Let
[tex]y=\pm \sqrt{x+2}[/tex]
[tex]\implies y^2-2=x[/tex]
x is defined for all y-values.
