Domain & Range of Inverse Function f-1: x≤0, y≤0

[FlopperJr]

Homework Statement

The function f:x→ 4-x² for the domain x≤0. Find the inverse of is denoted by f^-1 and state the domain and range of f^-1.

Homework Equations

Set equation to 0 and solve for x to find inverse. The D and R is going to be switched for the inverse..?

The Attempt at a Solution

I think I have found the inverse and range for this problem:
f^-1(x) = -√4-x

and has a range of y≤0 (which was given in problem)

not sure of range. Are my solutions correct?

 

[SammyS]

Homework Statement

The function f:x→ 4-x² for the domain x≤0. Find the inverse of is denoted by f^-1 and state the domain and range of f^-1.

Homework Equations

Set equation to 0 and solve for x to find inverse. The D and R is going to be switched for the inverse..?

The Attempt at a Solution

I think I have found the inverse and range for this problem:
f^-1(x) = -√4-x

and has a range of y≤0 (which was given in problem)

not sure of range. Are my solutions correct?

You should really put 4-x in parentheses (or use LaTeX):

f^-1(x) = -√(4-x)​

The Range of f^-1(x) is the same as the Domain of f(x). You are correct about this.

Do you know the Range of f(x) ?

 

[FlopperJr]

No, That is my problem. I tried but I keep getting wrong answer. 4-x^2=0
x=+/-2

 

[SammyS]

No, That is my problem. I tried but I keep getting wrong answer. 4-x^2=0
x=+/-2

How would you describe the graph of y = 4-x² ?

 

[FlopperJr]

How would you describe the graph of y = 4-x² ?

Its a parabola. It opens down, with the max point on 4.

 

[Mark44]

But you have a restricted domain, with x <= 0.

 

[FlopperJr]

I believe the domain and range of the inverse is
x>4 and y≤0

 

[Mark44]

The domain of f is {x | x <= 0}
The range of f is what? You know what the graph of f looks like, so you should be able to answer that by looking at the graph.

Then look at what SammyS said in post #2 about the domain and range of f^-1.

 

[FlopperJr]

Okay thanks! I know that it is going to be oppsite of whatever the original function was.
Now that I see the graph, it would be x≤4.

 

[Mark44]

That's better.

 

[FlopperJr]

Thanks so much for your help! I always have trouble with the domain and range.