Answer :
To find the inverse of the function [tex]\( f(x) = (x - 2)^3 \)[/tex], we will follow a step-by-step algebraic approach.
### Steps to Find the Inverse Function:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = (x - 2)^3 \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = (x - 2)^3 \][/tex]
3. To isolate [tex]\( x \)[/tex], take the cube root of both sides:
[tex]\[ \sqrt[3]{y} = x - 2 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt[3]{y} + 2 \][/tex]
5. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to write the inverse function:
[tex]\[ f^{-1}(x) = \sqrt[3]{x} + 2 \][/tex]
### Conclusion:
The inverse of the function [tex]\( f(x) = (x - 2)^3 \)[/tex] is:
[tex]\[ f^{-1}(x) = \sqrt[3]{x} + 2 \][/tex]
So the correct answer from the given options is:
[tex]\[ f^1(x) = \sqrt[3]{x} + 2 \][/tex]
### Verification
To ensure that this is correct, we can verify by composing [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex]:
- Compute [tex]\( f(f^{-1}(x)) \)[/tex]:
[tex]\[ f(f^{-1}(x)) = \left( \sqrt[3]{x} + 2 - 2 \right)^3 = (\sqrt[3]{x})^3 = x \][/tex]
- Compute [tex]\( f^{-1}(f(x)) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = \sqrt[3]{(x - 2)^3} + 2 - 2 = x - 2 + 2 = x \][/tex]
Both compositions return the original variable [tex]\( x \)[/tex], confirming that the functions are indeed inverses of each other.
### Steps to Find the Inverse Function:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = (x - 2)^3 \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = (x - 2)^3 \][/tex]
3. To isolate [tex]\( x \)[/tex], take the cube root of both sides:
[tex]\[ \sqrt[3]{y} = x - 2 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt[3]{y} + 2 \][/tex]
5. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to write the inverse function:
[tex]\[ f^{-1}(x) = \sqrt[3]{x} + 2 \][/tex]
### Conclusion:
The inverse of the function [tex]\( f(x) = (x - 2)^3 \)[/tex] is:
[tex]\[ f^{-1}(x) = \sqrt[3]{x} + 2 \][/tex]
So the correct answer from the given options is:
[tex]\[ f^1(x) = \sqrt[3]{x} + 2 \][/tex]
### Verification
To ensure that this is correct, we can verify by composing [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex]:
- Compute [tex]\( f(f^{-1}(x)) \)[/tex]:
[tex]\[ f(f^{-1}(x)) = \left( \sqrt[3]{x} + 2 - 2 \right)^3 = (\sqrt[3]{x})^3 = x \][/tex]
- Compute [tex]\( f^{-1}(f(x)) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = \sqrt[3]{(x - 2)^3} + 2 - 2 = x - 2 + 2 = x \][/tex]
Both compositions return the original variable [tex]\( x \)[/tex], confirming that the functions are indeed inverses of each other.
