To find the equation of the inverse relation for the given relation [tex]\( y = 4x - 8 \)[/tex], we need to follow a series of steps to determine the inverse function. Here is the detailed, step-by-step solution:
1. Start with the given equation:
[tex]\[
y = 4x - 8
\][/tex]
2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse:
This means swapping the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. The equation now becomes:
[tex]\[
x = 4y - 8
\][/tex]
3. Solve the new equation for [tex]\( y \)[/tex]:
Isolate [tex]\( y \)[/tex] on one side of the equation. To do this, we must perform algebraic operations to solve for [tex]\( y \)[/tex].
[tex]\[
x = 4y - 8
\][/tex]
Add 8 to both sides:
[tex]\[
x + 8 = 4y
\][/tex]
Divide both sides by 4:
[tex]\[
\frac{x + 8}{4} = y
\][/tex]
Simplify the equation:
[tex]\[
y = \frac{x}{4} + 2
\][/tex]
4. Write the inverse relation:
Now, replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to represent the inverse function:
[tex]\[
x = \frac{x}{4} + 2
\][/tex]
In conclusion, the equation of the inverse relation is:
[tex]\[
\boxed{x = \frac{x}{4} + 2}
\][/tex]
