To determine which function is the inverse of [tex]\( k(x) = 2x^5 + 6 \)[/tex], we need to find the inverse function [tex]\( k^{-1}(x) \)[/tex]. This involves the following steps:
1. Let [tex]\( y = k(x) \)[/tex]. Then:
[tex]\[
y = 2x^5 + 6
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 2y^5 + 6
\][/tex]
3. Subtract 6 from both sides:
[tex]\[
x - 6 = 2y^5
\][/tex]
4. Divide both sides by 2:
[tex]\[
\frac{x - 6}{2} = y^5
\][/tex]
5. Take the fifth root of both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt[5]{\frac{x - 6}{2}}
\][/tex]
So, the inverse function is [tex]\( k^{-1}(x) = \sqrt[5]{\frac{x - 6}{2}} \)[/tex].
Now, let's compare this result with the given options:
1. [tex]\( h(x) = \sqrt[5]{\frac{x - 6}{2}} \)[/tex]
2. [tex]\( h(x) = \sqrt[5]{\frac{x + 6}{2}} \)[/tex]
3. [tex]\( h(x) = \frac{\sqrt[5]{x}}{2} - 6 \)[/tex]
4. [tex]\( h(x) = \sqrt[5]{\frac{x}{2}} - 6 \)[/tex]
The correct inverse function [tex]\( k^{-1}(x) \)[/tex] matches the first option:
[tex]\[
h(x) = \sqrt[5]{\frac{x - 6}{2}}
\][/tex]
Therefore, the function that is the inverse of [tex]\( k(x) = 2x^5 + 6 \)[/tex] is:
[tex]\[
\boxed{h(x) = \sqrt[5]{\frac{x - 6}{2}}}
\][/tex]
