To algebraically 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) = -1 ±sqrt(x)/2
Result: No, the inverse is not 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 will switch x and y and solve for y.
y=(2 x+1)^2 → x=(2 y+1)^2
The resulting equation will be the inverse of the given function.
Finally, to indicate that this is the inverse function of f(x), we replace y with f^(- 1)(x).
f^(- 1)(x)= -1±sqrt(x)/2
Now that we have 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
The domain of the given function is the set of all real numbers.
Domain:& x∈R
For the range, think about the fact that the square of a number is always non-negative.
f(x)=(2x+1)^2 ≥ 0
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 the 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∈R
Is the Inverse a Function?
A function is a relation where each input is related to exactly one output. In our case, we can see that for each x in the domain of f^(- 1)(x), there are two values of y.
If x=a, then y=-1±sqrt(a)/2.
Therefore, the inverse of the function is not a function.
