Question

Find the inverse of k(x)=x²-3. Define the restrictions to the domain.

Answer 1

Inverse function:
`\pm √(x + 3)`

Domain:
`x \ge -3`

Explanation:

`k(x) = x^2-3`

To find the inverse, set y = k(x)

==> y = x² - 3

Swap x and y

x = y² - 3

Add 3 to both sides
x + 3 = y²

y² = x + 3

`y = \pm √(x + 3)`

or, in other words:

`y = √(x +3) \:\;and\; y = -√(x + 3)` are the two solutions

Replace Swap x and y again to get the inverse as a function of x

`k^(-1)(x) = √(x+3),\:-√(x+3)`

Since square roots of negative numbers are not defined, the term under the square root must be ≥ 0

x + 3 ≥ 0

==> x ≥ -3

There is no restriction on the upper bound it can go upto infinity

Domain of
`√(x + 3) : \:x\ge \:-3`

Answer 2

Answer: The inverse of the function k(x)=x²-3 is k^(-1)(x)=±√(x+3). The domain restrictions for the inverse are x≥-3, since the square root is only defined for non-negative values.

The inverse of a function is the reflection of the original function over the line y=x. To find the inverse of k(x)=x²-3, we first swap the x and y variables: y=x²-3. Then, we solve for x: x=±√(y+3). This gives us the inverse function k^(-1)(x)=±√(x+3).

However, it is important to note that the square root function is only defined for non-negative values. Therefore, we must restrict the domain of the inverse function to x≥-3. This means that the inverse function can only be applied to input values that are greater than or equal to -3. For values less than -3, the inverse function is undefined.

Explanation: