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\}.
