Which of the following is the inverse of the function [tex][tex]$k(x) = 2x^5 + 6$[/tex][/tex]?

A. [tex][tex]$h(x) = \sqrt[5]{\frac{x-6}{2}}$[/tex][/tex]
B. [tex][tex]$h(x) = \sqrt[5]{\frac{x+6}{2}}$[/tex][/tex]
C. [tex][tex]$h(x) = \frac{\sqrt[5]{x}}{2} - 6$[/tex][/tex]
D. [tex][tex]$h(x) = \sqrt[5]{\frac{x}{2}} - 6$[/tex][/tex]



Answer :

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]