select the correct answer.\nwhich function is the inverse of $f(x) = -x^3 - 9$?\na. $f^{-1}(x) = \\sqrt3{x +…

Explanation:

Step 1: Replace ( f(x) ) with ( y )

We start with the function ( f(x) = -x^3 - 9 ). Replace ( f(x) ) with ( y ), so we have ( y = -x^3 - 9 ).

Step 2: Swap ( x ) and ( y )

To find the inverse, we swap the roles of ( x ) and ( y ). This gives us ( x = -y^3 - 9 ).

Step 3: Solve for ( y )

First, we add 9 to both sides of the equation: ( x + 9 = -y^3 ) Then, we multiply both sides by -1: ( -x - 9 = y^3 ) Or, equivalently, ( y^3 = -x - 9 ) To solve for ( y ), we take the cube root of both sides: ( y = \sqrt[3]{-x - 9} ) Wait, no, let's check again. Wait, from ( x = -y^3 - 9 ), let's solve for ( y ) correctly.

Starting over from ( x = -y^3 - 9 ):

Add 9 to both sides: ( x + 9 = -y^3 )

Multiply both sides by -1: ( -x - 9 = y^3 )

Take cube root: ( y = \sqrt[3]{-x - 9} )? Wait, no, that's not one of the options. Wait, maybe I made a mistake.

Wait, original function: ( y = -x^3 - 9 )

Swap ( x ) and ( y ): ( x = -y^3 - 9 )

Let's solve for ( y ):

Add 9 to both sides: ( x + 9 = -y^3 )

Multiply both sides by -1: ( -x - 9 = y^3 )? No, wait, ( x + 9 = -y^3 ) => ( y^3 = - (x + 9) = -x - 9 )? No, that's not right. Wait, ( x + 9 = -y^3 ) => ( y^3 = -x - 9 )? Wait, no, ( x + 9 = -y^3 ) => ( y^3 = - (x + 9) = -x - 9 ). Then ( y = \sqrt[3]{-x - 9} ). But that's option B? Wait, no, option B is ( \sqrt[3]{-x - 9} ). But let's check the options again.

Wait, maybe I made a mistake in the algebra. Let's try again.

Original function: ( y = -x^3 - 9 )

Swap ( x ) and ( y ): ( x = -y^3 - 9 )

Solve for ( y ):

Add 9 to both sides: ( x + 9 = -y^3 )

Multiply both sides by -1: ( -x - 9 = y^3 )

Take cube root: ( y = \sqrt[3]{-x - 9} ). So that's option B? But wait, let's check the options again.

Wait, option B is ( f^{-1}(x) = \sqrt[3]{-x - 9} ). But let's check the options again. Wait, maybe I made a mistake. Wait, let's check the options:

A. ( \sqrt[3]{x + 9} )

B. ( \sqrt[3]{-x - 9} )

C. ( -\sqrt[3]{-x + 9} )

D. ( -\sqrt[3]{x - 9} )

Wait, maybe my algebra is wrong. Let's try another approach. Let's take the original function ( f(x) = -x^3 - 9 ). Let's find ( f(f^{-1}(x)) ) for option C. Wait, maybe I made a mistake in the sign.

Wait, let's try option C: ( f^{-1}(x) = -\sqrt[3]{-x + 9} )

Let's compute ( f(f^{-1}(x)) ):

( f(f^{-1}(x)) = - (f^{-1}(x))^3 - 9 )

Substitute ( f^{-1}(x) = -\sqrt[3]{-x + 9} ):

( (f^{-1}(x))^3 = \left( -\sqrt[3]{-x + 9} \right)^3 = - (-x + 9) = x - 9 )

Then ( f(f^{-1}(x)) = - (x - 9) - 9 = -x + 9 - 9 = -x ). No, that's not x. So that's wrong.

Wait, let's try option C again. Wait, maybe I messed up.

Wait, let's try option C: ( f^{-1}(x) = -\sqrt[3]{-x + 9} )

Compute ( f(f^{-1}(x)) ):

( f(f^{-1}(x)) = - \left( f^{-1}(x) \right)^3 - 9 )

( \left( f^{-1}(x) \right)^3 = \left( -\sqrt[3]{-x + 9} \right)^3 = - (-x + 9) = x - 9 )

So ( f(f^{-1}(x)) = - (x - 9) - 9 = -x + 9 - 9 = -x ). Not x. So that's wrong.

Wait, let's try option B: ( f^{-1}(x) = \sqrt[3]{-x - 9} )

Compute ( f(f^{-1}(x)) ):

( f(f^{-1}(x)) = - \left( \sqrt[3]{-x - 9} \right)^3 - 9 = - (-x - 9) - 9 = x + 9 - 9 = x ). Ah, that works.

Wait, but let's check the options again. Option B is ( \sqrt[3]{-x - 9} ). But let's check the original problem again.

Wait, maybe I made a mistake in the algebra. Let's do it step by step.

Original function: ( y = -x^3 - 9 )

Swap x and y: ( x = -y^3 - 9 )

Solve for y:

Add 9 to both sides: ( x + 9 = -y^3 )

Multiply both sides by -1: ( -x - 9 = y^3 )

Take cube root: ( y = \sqrt[3]{-x - 9} )

Which is option B. So the correct answer is B.

Wait, but let's check the options again. Option B is ( f^{-1}(x) = \sqrt[3]{-x - 9} ). So that's the inverse.

But wait, let's verify with a value. Let's take x = 0 in the original function. Then f(0) = -0 - 9 = -9. So the inverse function should take -9 to 0. Let's check option B: f^{-1}(-9) = \sqrt[3]{-(-9) - 9} = \sqrt[3]{9 - 9} = \sqrt[3]{0} = 0. Correct.

Another test: take x = 1 in original function: f(1) = -1 - 9 = -10. Then inverse function should take -10 to 1. Option B: f^{-1}(-10) = \sqrt[3]{-(-10) - 9} = \sqrt[3]{10 - 9} = \sqrt[3]{1} = 1. Correct.

So the correct answer is B.

Answer:

B. ( f^{-1}(x) = \sqrt[3]{-x - 9} )