Answered

Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.

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\}.
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.