Answer :
We start with the function
[tex]$$
h(x) = 6x^7 - 9.
$$[/tex]
To find the inverse, we let
[tex]$$
y = 6x^7 - 9,
$$[/tex]
and solve for [tex]$x$[/tex] in terms of [tex]$y$[/tex].
1. Add [tex]$9$[/tex] to both sides:
[tex]$$
y + 9 = 6x^7.
$$[/tex]
2. Divide both sides by [tex]$6$[/tex]:
[tex]$$
\frac{y + 9}{6} = x^7.
$$[/tex]
3. Take the seventh root of both sides to solve for [tex]$x$[/tex]:
[tex]$$
x = \left(\frac{y + 9}{6}\right)^{\frac{1}{7}}.
$$[/tex]
Since [tex]$x = g(y)$[/tex] when the function is inverted, the inverse function is
[tex]$$
g(y) = \left(\frac{y + 9}{6}\right)^{\frac{1}{7}}.
$$[/tex]
Thus, the final answer is
[tex]$$
\boxed{g(y)=\left(\frac{y+9}{6}\right)^{\frac{1}{7}}}.
$$[/tex]
[tex]$$
h(x) = 6x^7 - 9.
$$[/tex]
To find the inverse, we let
[tex]$$
y = 6x^7 - 9,
$$[/tex]
and solve for [tex]$x$[/tex] in terms of [tex]$y$[/tex].
1. Add [tex]$9$[/tex] to both sides:
[tex]$$
y + 9 = 6x^7.
$$[/tex]
2. Divide both sides by [tex]$6$[/tex]:
[tex]$$
\frac{y + 9}{6} = x^7.
$$[/tex]
3. Take the seventh root of both sides to solve for [tex]$x$[/tex]:
[tex]$$
x = \left(\frac{y + 9}{6}\right)^{\frac{1}{7}}.
$$[/tex]
Since [tex]$x = g(y)$[/tex] when the function is inverted, the inverse function is
[tex]$$
g(y) = \left(\frac{y + 9}{6}\right)^{\frac{1}{7}}.
$$[/tex]
Thus, the final answer is
[tex]$$
\boxed{g(y)=\left(\frac{y+9}{6}\right)^{\frac{1}{7}}}.
$$[/tex]
