Question

Please help! functions operations. explain please

Answer 1

f(x)

Domain: all real numbers

Range:
`y\ge -2`

`f^(-1)(x)`

Domain:
`y\ge -2`

Range:all real numbers

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 √(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 √(x+2)=y`

`f^(-1)(x)=\pm √(x+2)`

Domain:
`x\ge-2`

Range:

Let

`y=\pm √(x+2)`

`\implies y^2-2=x`

x is defined for all y-values.