To determine whether the function [tex]\( f(x) = \sqrt[3]{4x} \)[/tex] is one-to-one, we need to verify if each output value corresponds to exactly one input value. A function is one-to-one if and only if [tex]\( f(a) = f(b) \)[/tex] implies [tex]\( a = b \)[/tex] for any given [tex]\( a \)[/tex] and [tex]\( b \)[/tex] in the domain of [tex]\( f \)[/tex].
### Step 1: Determine if the function is one-to-one
Consider the function [tex]\( f(x) = \sqrt[3]{4x} \)[/tex].
- Let [tex]\( f(a) = f(b) \)[/tex].
- Then, [tex]\( \sqrt[3]{4a} = \sqrt[3]{4b} \)[/tex].
Since the cube root function is one-to-one, the equality [tex]\( \sqrt[3]{4a} = \sqrt[3]{4b} \)[/tex] implies [tex]\( 4a = 4b \)[/tex].
- Dividing both sides by 4, we get [tex]\( a = b \)[/tex].
Therefore, the function [tex]\( f(x) = \sqrt[3]{4x} \)[/tex] is one-to-one.
### Step 2: Find the inverse function
To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we need to solve the equation [tex]\( y = \sqrt[3]{4x} \)[/tex] for [tex]\( x \)[/tex].
1. Start with the equation:
[tex]\[ y = \sqrt[3]{4x} \][/tex]
2. Cube both sides to get rid of the cube root:
[tex]\[ y^3 = 4x \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{y^3}{4} \][/tex]
### Conclusion:
Since the function is one-to-one, we can find its inverse. The inverse function is:
[tex]\[ f^{-1}(x) = \frac{x^3}{4} \][/tex]
Therefore, the correct choice is:
A. Yes, the function is one-to-one. The inverse function is [tex]\( f^{-1}(x) = \frac{x^3}{4} \)[/tex].
