To find the inverse of the function [tex]\( g(x) = 8x + 3 \)[/tex], you can follow these steps:
1. Express the function in terms of [tex]\( y \)[/tex]:
[tex]\[
y = 8x + 3
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to set up for inverting the function:
[tex]\[
x = 8y + 3
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
To isolate [tex]\( y \)[/tex], first subtract the constant term from both sides:
[tex]\[
x - 3 = 8y
\][/tex]
Then, divide both sides by the coefficient of [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x - 3}{8}
\][/tex]
4. Simplify the expression:
We can further simplify the above expression:
[tex]\[
y = \frac{x}{8} - \frac{3}{8}
\][/tex]
5. Rewrite the inverse function:
The inverse function [tex]\( g^{-1}(x) \)[/tex] is written in terms of [tex]\( x \)[/tex]:
[tex]\[
g^{-1}(x) = \frac{x}{8} - \frac{3}{8}
\][/tex]
Therefore, the simplified form of the inverse function is:
[tex]\[
g^{-1}(x) = \frac{x}{8} - \frac{3}{8}
\][/tex]
