Final answer:
The inverse function of f(x) = x^2 + 7 can be found by swapping x and y and solving for y, but we have to restrict the domain of x^2 to make the function one-to-one. If we assume the domain is restricted to x ≥ 0, the inverse function is f^-1(x) = √(x - 7), with x ≥ 7.
Step-by-step explanation:
The inverse of a function f(x) = x^2 + 7 is a function that 'undoes' the original function.
To find the inverse, you typically switch x and y and then solve for y.
In this case, however, we must be cautious, because x^2 is not a one-to-one function (it does not pass the horizontal line test) unless we restrict its domain.
For example, the inverse of y = x^2 is x = √(y) or x = -√(y), but this only holds true when we define the domain of the original function such as x ≥ 0 or x < 0.
To formally express the inverse, assuming we have restricted the domain of the original function so it is one-to-one (let's say x ≥ 0 to have a real function), we start by writing y = x^2 + 7.
Then we swap x and y to get x = y^2 + 7 and solve for y.
To do this, we subtract 7 from both sides to get x - 7 = y^2, and then take the square root: √(x - 7) = y, which gives us our inverse function f-1(x) = √(x - 7), assuming x ≥ 7 to keep the square root defined in the real number system.