To find the inverse function [tex]\( g(y) \)[/tex] of [tex]\( h(x) = 3x^4 - 10 \)[/tex], we will follow these steps:
1. Start with the given function:
[tex]\[
h(x) = 3x^4 - 10
\][/tex]
2. Replace [tex]\( h(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = 3x^4 - 10
\][/tex]
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = 3x^4 - 10
\][/tex]
Add 10 to both sides:
[tex]\[
y + 10 = 3x^4
\][/tex]
Divide both sides by 3:
[tex]\[
\frac{y + 10}{3} = x^4
\][/tex]
To solve for [tex]\( x \)[/tex], take the fourth root of both sides. Note that [tex]\( x \)[/tex] can be positive or negative, but we often take the principal (positive) root in many contexts:
[tex]\[
x = \pm \left( \frac{y + 10}{3} \right)^{\frac{1}{4}}
\][/tex]
In a typical case where we take the principal root, we consider the positive root or sometimes the imaginary root, particularly if the context (like complex functions) permits. In this case:
4. Express the inverse function [tex]\( g(y) \)[/tex]:
[tex]\[
g(y) = -i \left( \frac{y + 10}{3} \right)^{\frac{1}{4}}
\][/tex]
Hence, the inverse function is:
[tex]\[
g(y) = -i\left( \frac{y + 10}{3} \right)^{\frac{1}{4}}
\][/tex]
