Answer
The inverse of this function is given as
f⁻¹(x) = √(x - 10)
Domain in interval form = [10, ∞)
OR
Domain in inequality form = 10 ≤ x < ∞
Step-by-step explanation
First of, the inverse of a function is a function that reverses the actions of the function. That is, for the inverse function, if we are given f(x), we would be able to obtain x.
The step to finding a function's inverse is to write y instead of f(x). We then rewrite by replacing the y by x and then make y the subject of formula, the y which is a subject of formula now, represents the inverse function, f⁻¹(x).
f(x) = x² + 10
We will write y instead of f(x)
y = x² + 10
We then replace y by x
x = y² + 10
We will then make y the subject of formula
x = y² + 10
y² + 10 = x
y² = x - 10
Take the square root of both sides
√(y²) = √(x - 10)
y = √(x - 10)
f⁻¹(x) = √(x - 10)
For the domain of this inverse of this function, we know that there is no real solution for the square root of a negative number, we know that the domain of this function has to be values of x that make the expression under the square root equal to or greater than 0.
x - 10 ≥ 0
Add 10 to both sides
x - 10 + 10 ≥ 0 + 10
x ≥ 10
Domain = [10, ∞) OR 10 ≤ x < ∞
Hope this Helps!!!