Let's solve each part step-by-step.
### Part (a): Find the inverse function [tex]\( f^{-1} \)[/tex]
Given the function [tex]\( f(x) = x^3 + 3 \)[/tex], we want to find its inverse [tex]\( f^{-1}(x) \)[/tex].
1. To find the inverse, we start by setting [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = x^3 + 3 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to solve for [tex]\( y \)[/tex]:
[tex]\[ x = y^3 + 3 \][/tex]
3. Isolate [tex]\( y \)[/tex]:
[tex]\[ x - 3 = y^3 \][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = (x - 3)^{\frac{1}{3}} \][/tex]
So the inverse function is:
[tex]\[ f^{-1}(x) = (x - 3)^{\frac{1}{3}} \][/tex]
The inverse function is valid for all [tex]\( x \)[/tex] since the cubic root function is defined for all real numbers. Thus, the correct choice is:
[tex]\[ \boxed{D. \ f^{-1}(x) = (x-3)^{\frac{1}{3}}, \ \text{for all} \ x} \][/tex]
### Part (b): Verify the inverse function
We need to verify two things:
1. [tex]\( f(f^{-1}(x)) = x \)[/tex]
2. [tex]\( f^{-1}(f(x)) = x \)[/tex]
Let's verify both:
#### Verification 1: [tex]\( f(f^{-1}(x)) = x \)[/tex]
1. Start with [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = (x - 3)^{\frac{1}{3}} \][/tex]
2. Substitute [tex]\( f^{-1}(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(f^{-1}(x)) = f((x - 3)^{\frac{1}{3}}) \][/tex]
3. Evaluate [tex]\( f \)[/tex]:
[tex]\[ f((x - 3)^{\frac{1}{3}}) = \left( (x - 3)^{\frac{1}{3}} \right)^3 + 3 \][/tex]
[tex]\[ = x - 3 + 3 \][/tex]
[tex]\[ = x \][/tex]
Thus, [tex]\( f(f^{-1}(x)) = x \)[/tex].
#### Verification 2: [tex]\( f^{-1}(f(x)) = x \)[/tex]
1. Start with [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x^3 + 3 \][/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( f^{-1} \)[/tex]:
[tex]\[ f^{-1}(f(x)) = f^{-1}(x^3 + 3) \][/tex]
3. Evaluate [tex]\( f^{-1} \)[/tex]:
[tex]\[ f^{-1}(x^3 + 3) = \left( (x^3 + 3) - 3 \right)^{\frac{1}{3}} \][/tex]
[tex]\[ = (x^3)^{\frac{1}{3}} \][/tex]
[tex]\[ = x \][/tex]
Thus, [tex]\( f^{-1}(f(x)) = x \)[/tex].
### Conclusion on verification
Since both verifications are correct, we've shown that [tex]\( f(f^{-1}(x)) = x \)[/tex] and [tex]\( f^{-1}(f(x)) = x \)[/tex]. Therefore, our inverse function [tex]\( f^{-1}(x) = (x-3)^{\frac{1}{3}} \)[/tex] is indeed correct.
