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The function [tex]$f(x)=x^{\frac{3}{7}}-10, x \geq 0$[/tex] is one-to-one. Use this information to answer parts (a) through (d).

(a) [tex]f^{-1}(x) = \square \, \square[/tex]
(Simplify your answer.)

(b) Find the domain of [tex]f[/tex]. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- A. The domain is [tex]\{x \mid x \neq \square \}[/tex].
- B. The domain is [tex]\{x \mid x \leq \square \}[/tex].
- C. The domain is [tex]\{x \mid x \geq \square \}[/tex].
- D. The domain is the set of all real numbers.

(c) Find the range of [tex]f[/tex]. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- A. The range is [tex]\{y \mid y \leq \square \}[/tex].
- B. The range is [tex]\{y \mid y \neq \square \}[/tex].
- C. The range is [tex]\{y \mid y \geq \square \}[/tex].
- D. The range is the set of all real numbers.

(d) Find the domain of [tex]f^{-1}[/tex]. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- A. The domain is [tex]\{x \mid x \neq \square \}[/tex].
- B. The domain is [tex]\{x \mid x \leq \square \}[/tex].
- C. The domain is [tex]\{x \mid x \geq \square \}[/tex].
- D. The domain is the set of all real numbers.

(e) Find the range of [tex]f^{-1}[/tex]. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- A. The range is [tex]\{y \mid y \leq \square \}[/tex].
- B. The range is [tex]\{y \mid y \geq \square \}[/tex].
- C. The range is [tex]\{y \mid y \neq \square \}[/tex].
- D. The range is the set of all real numbers.

Sagot :

GinnyAnswer
Let's go through this one step at a time.

### Part (a) - Finding the Inverse Function

Given:
[tex]\[ f(x) = x^{\frac{3}{7}} - 10 \][/tex]

We need to find the inverse function [tex]\( f^{-1}(x) \)[/tex].

1. Start with [tex]\( y = f(x) \)[/tex]:

[tex]\[ y = x^{\frac{3}{7}} - 10 \][/tex]

2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:

[tex]\[ y + 10 = x^{\frac{3}{7}} \][/tex]

3. Raise both sides to the power of [tex]\( \frac{7}{3} \)[/tex]:

[tex]\[ (y + 10)^{\frac{7}{3}} = x \][/tex]

Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:

[tex]\[ f^{-1}(x) = (x + 10)^{\frac{7}{3}} \][/tex]

### Part (b) - Finding the Domain and Range

#### 1. Domain of [tex]\( f \)[/tex]

We are given [tex]\( x \geq 0 \)[/tex]. Hence, the domain of [tex]\( f \)[/tex] is:
[tex]\[ \{x \mid x \geq 0\} \][/tex]

So, the correct choice is:
C. The domain is \{x \mid x \geq 0\}.

#### 2. Range of [tex]\( f \)[/tex]

As [tex]\( x \)[/tex] goes from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 0^{\frac{3}{7}} - 10 = -10 \)[/tex]
- As [tex]\( x \to \infty \)[/tex], [tex]\( x^{\frac{3}{7}} \to \infty \)[/tex] and thus [tex]\( f(x) \to \infty - 10 \)[/tex], meaning [tex]\( f(x) \to \infty \)[/tex]

Therefore, [tex]\( f(x) \)[/tex] ranges from [tex]\( -10 \)[/tex] to [tex]\( \infty \)[/tex]. The range of [tex]\( f \)[/tex] is:
[tex]\[ \{y \mid y \geq -10\} \][/tex]

So, the correct choice is:
C. The range is \{y \mid y \geq -10\}.

#### 3. Domain of [tex]\( f^{-1} \)[/tex]

The domain of the inverse function [tex]\( f^{-1}(x) \)[/tex] is the same as the range of [tex]\( f(x) \)[/tex]. Therefore, the domain of [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ \{x \mid x \geq -10\} \][/tex]

So, the correct choice is:
C. The domain is \{x \mid x \geq -10\}.

#### 4. Range of [tex]\( f^{-1} \)[/tex]

The range of the inverse function [tex]\( f^{-1}(x) \)[/tex] is the same as the domain of [tex]\( f(x) \)[/tex]. Therefore, the range of [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ \{y \mid y \geq 0\} \][/tex]

So, the correct choice is:
C. The range is \{y \mid y \geq 0\}.

### Summary of Answers

(a) [tex]\( f^{-1}(x) = (x + 10)^{\frac{7}{3}} \)[/tex]

(b)
- Domain of [tex]\( f \)[/tex]: C. The domain is \{x \mid x \geq 0\}.
- Range of [tex]\( f \)[/tex]: C. The range is \{y \mid y \geq -10\}.
- Domain of [tex]\( f^{-1} \)[/tex]: C. The domain is \{x \mid x \geq -10\}.
- Range of [tex]\( f^{-1} \)[/tex]: C. The range is \{y \mid y \geq 0\}.
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