How do you find the inverse of f(x)=sqrt(x-3)?

2 Answers
May 20, 2017

f^-1(x)=x^2+3 where x>=0

Explanation:

You can find the inverse function by exchanging x and y and then solve for y in terms of x.

y=sqrt(x-3) (Original function)
x=sqrt(y-3)
x^2=y-3
x^2+3=y
Therefore
f^-1(x)=x^2+3

The domain of the inverse function, f^-1(x) is equal to the range of the original function f(x). The range should be [0, +oo) if the domain of f(x) is [3, +oo).

May 20, 2017

The inverse is f^-1(x)=x^2+3

Explanation:

The domain of f(x) is x in [3,+oo) and the range is y in [0,+oo)

Let color(blue)(y)=sqrt(color(red)(x)-3)

Interchange x and y

color(red)(x)=sqrt(color(blue)(y)-3)

Now, we express y in terms of x

x^2=y-3

y=x^2+3

Therefore,

f^-1(x)=x^2+3

The domain and range of f^-1(x) is the range and domain of f(x)

The domain of f^-1(x) is x in [0,+oo) and the range is y in [3,+oo)

Verification

f(f^-1(x))=f(x^2+3)=sqrt(x^2+3-3)=x