To determine the inverse of f(x), first replace f(x) with y. Then switch x and y and solve for y.
Inverse: f^(-1)(x) = (x+4)^2 - 3, x≥-4 Result: Yes, the inverse is a function.
Practice makes perfect
We will begin by finding the inverse of f(x). First, we need to replace f(x) with y. From there, we switch x and y and solve for y.
y=sqrt(x+3)-4 → x=sqrt(y+3)-4
The resulting equation will be the inverse of the given function.
All values of x that are greater than or equal to -3 are included in the domain.
Domain: x≥-3
For the range, think about the fact that the principal root of a number is always positive.
lf(x) l= lsqrt(x+3) ≥0 l-4
Because the f(x) is difference of non-negative number and 4, the range is all values of y such that y≥-4.
Range:& y≥-4
Domain and Range of the Inverse
When we find the inverse of a function, we are basically exchanging its x and y values. Therefore, the range of f(x) becomes the domain of f^(- 1)(x), and the domain of f(x) becomes the range of f^(- 1)(x).
Domain: x ≥ -4
Range: y ≥ -3
Is the Inverse a Function?
A function is a relation where each input is related to exactly one output. In this case, for each x in domain of f^(- 1)(x), there is only one value of y in the range. Therefore, f^(- 1)(x) is a function in its domain.
