Question

question 4 of 10, step 1 of 1 determine if the following function has an inverse function. \\( a(x) = 3sqrt{x^5} \\) answer \\( \bigcirc \\) a has an inverse function \\( \bigcirc \\) a does not have an inverse function

Explanation:

Step1: Rewrite the function

$A(x) = 3x^{\frac{5}{2}}$

Step2: Check one-to-one property

The function $A(x) = 3x^{\frac{5}{2}}$ is defined for $x\geq0$. Its derivative is $A'(x) = 3\times\frac{5}{2}x^{\frac{3}{2}} = \frac{15}{2}x^{\frac{3}{2}}$, which is non-negative for all $x\geq0$, and positive for $x>0$. This means the function is strictly increasing on its domain, so it is one-to-one.

Step3: Invertibility of one-to-one functions

A function has an inverse if and only if it is one-to-one.

Answer:

A has an inverse function