To determine the inverse of f(x), first replace f(x) with y. Then switch x and y and solve for y.
Function
Inverse of the Function
Equation
f(x)=1/sqrt(-2x)
f^(- 1)(x)=-1/2x^2
Domain
x<0
x > 0
Range
y > 0
y < 0
Result: 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=1/sqrt(-2 x) → x=1/sqrt(-2 y)
The resulting equation will be the inverse of the given function.
Finally, to indicate that this is the inverse function of f(x), we will replace y with f^(- 1)(x).
f^(- 1)(x)=-1/2x^2
Now that we found the inverse of f(x), we will determine the domain and range of f(x) and f^(-1).
Domain and Range of the Function
To determine the domain of the given function, the radicand cannot be negative and the denominator of the quotient cannot be 0. This means that the radicand must be greater than 0. Let's find the value(s) of x that make this true.
All values of x that are less than 0 are included in the domain.
Domain: x< 0
For the range, think about the fact that the principal root of a number is always positive.
f(x)=1/sqrt(-2x) l← ← lpositive positive
Because the quotient of two positive numbers is always a positive number, the range is all values of y such that y>0.
Range:& y>0
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 > 0
Range: y < 0
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.
