To find the inverse function [tex]\( f^{-1}(x) \)[/tex] of the given function [tex]\( f(x) = x^3 - 7 \)[/tex], we need to follow several steps to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] where [tex]\( y = f(x) \)[/tex]. Here is the step-by-step process:
1. Start with the function definition:
[tex]\[ y = f(x) = x^3 - 7 \][/tex]
2. Switch the roles of [tex]\( y \)[/tex] and [tex]\( x \)[/tex] to find the inverse:
[tex]\[ x = y^3 - 7 \][/tex]
3. Isolate the term containing [tex]\( y \)[/tex]:
[tex]\[ y = x^3 - 7 \][/tex]
4. Solve for [tex]\( y \)[/tex]:
- Add 7 to both sides:
[tex]\[ y + 7 = x^3 \][/tex]
5. Take the cube root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt[3]{y + 7} \][/tex]
6. Rewrite [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] to get the inverse function:
[tex]\[ x = \sqrt[3]{y + 7} \][/tex]
Now, replacing [tex]\( y \)[/tex] back with [tex]\( x \)[/tex], we have the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = (x + 7)^{1/3} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = (x + 7)^{1/3} \][/tex]
