Question

Please help! functions operations. explain please

Answer 1

Answer:

f(x)

Domain: all real numbers

Range:$y\ge -2$

$f^{-1}(x)$

Domain: $y\ge -2$

Range:all real numbers

Step-by-step explanation:

Given $f(x)=x^2-2$

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 $y=x^2-2$

Solve for x;

$x=\pm \sqrt{y+2}$

This function is defined for $y+2\ge0$

This implies that; $y\ge-2$

Inverse Function

The domain of f(x) becomes the range of $f^{-1}(x)$ and the range of  f(x) becomes the domain of $f^{-1}(x)$.

Domain: $y\ge -2$

Range:all real numbers

Or see it in details

Let $y=x^2-2$

interchange x and y;

$x=y^2-2$

$x+2=y^2$

$\pm \sqrt{x+2}=y$

$f^{-1}(x)=\pm \sqrt{x+2}$

Domain: $x\ge-2$

Range:

Let

$y=\pm \sqrt{x+2}$

$\implies y^2-2=x$

x is defined for all y-values.