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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][tex]$f^{-1}(f(x)) = x$[/tex][/tex].

a. Select the correct choice below and fill in the answer box(es) to complete your choice.
(Simplify your answer. Use integers or fractions for any numbers in the expression.)

A. [tex]$f^{-1}(x) = \square$[/tex], for [tex]$x \leq \square$[/tex]
B. [tex][tex]$f^{-1}(x) = \square$[/tex][/tex], for [tex]$x \geq \square$[/tex]
C. [tex]$f^{-1}(x) = \square$[/tex], for [tex][tex]$x \neq \square$[/tex][/tex]
D. [tex]$f^{-1}(x) = (x - 3)^{\frac{1}{3}}$[/tex], for all [tex]$x$[/tex]

b. Verify that the equation is correct.

[tex]
\begin{array}{l}
f\left(f^{-1}(x)\right) = \\
\text{and} \, f^{-1}(f(x)) = \\
\end{array}
[/tex]

Substitute.
Simplify.

Sagot :

GinnyAnswer
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.
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