Let's address the problem step-by-step:
### Part (a): Finding the Inverse Function of [tex]\( f(x) \)[/tex]
The given function is:
[tex]\[ f(x) = x^3 + 3 \][/tex]
To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we start by setting:
[tex]\[ y = f(x) \][/tex]
[tex]\[ y = x^3 + 3 \][/tex]
We need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. To do this, we solve for [tex]\( x \)[/tex]:
[tex]\[ y = x^3 + 3 \][/tex]
Subtract 3 from both sides:
[tex]\[ y - 3 = x^3 \][/tex]
Now take the cube root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = (y - 3)^{1/3} \][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = (x - 3)^{1/3} \][/tex]
### Part (b): Verifying the Inverse Function
We need to verify two properties:
1. [tex]\( f(f^{-1}(x)) = x \)[/tex]
2. [tex]\( f^{-1}(f(x)) = x \)[/tex]
#### Verifying [tex]\( f(f^{-1}(x)) = x \)[/tex]
First, substitute [tex]\( f^{-1}(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(f^{-1}(x)) = f((x - 3)^{1/3}) \][/tex]
Using the original function [tex]\( f \)[/tex]:
[tex]\[ f((x - 3)^{1/3}) = ((x - 3)^{1/3})^3 + 3 \][/tex]
Simplifying,
[tex]\[ ((x - 3)^{1/3})^3 = x - 3 \][/tex]
So,
[tex]\[ f((x - 3)^{1/3}) = x - 3 + 3 \][/tex]
[tex]\[ f((x - 3)^{1/3}) = x \][/tex]
Thus,
[tex]\[ f(f^{-1}(x)) = x \][/tex]
#### Verifying [tex]\( f^{-1}(f(x)) = x \)[/tex]
Next, substitute [tex]\( f(x) \)[/tex] into [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = f^{-1}(x^3 + 3) \][/tex]
Using the inverse function [tex]\( f^{-1} \)[/tex]:
[tex]\[ f^{-1}(x^3 + 3) = (x^3 + 3 - 3)^{1/3} \][/tex]
[tex]\[ f^{-1}(x^3 + 3) = (x^3)^{1/3} \][/tex]
Simplifying,
[tex]\[ (x^3)^{1/3} = x \][/tex]
Thus,
[tex]\[ f^{-1}(f(x)) = x \][/tex]
### Conclusion
We've found that the inverse function of [tex]\( f(x) = x^3 + 3 \)[/tex] is:
[tex]\[ f^{-1}(x) = (x - 3)^{1/3} \][/tex]
Additionally, we verified that [tex]\( f(f^{-1}(x)) = x \)[/tex] and [tex]\( f^{-1}(f(x)) = x \)[/tex], confirming that our inverse function is correct.
