The function [tex]f(x) = x^3 + 3[/tex] is one-to-one.

a. Find an equation for [tex]f^{-1}[/tex], the inverse function.

b. Verify that your equation is correct by showing that [tex]f\left(f^{-1}(x)\right) = x[/tex] and [tex]f^{-1}(f(x)) = x[/tex].



Answer :

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.