Given g(x) = ³√((1 + 4x)/(6 + x)), find g⁻¹(x)

Step 1: Let y = g(x)
y = ³√((1 + 4x)/(6 + x))

Step 2: Swap x and y
x = ³√((1 + 4y)/(6 + y))

Step 3: Cube both sides
x³ = (1 + 4y)/(6 + y)

Step 4: Cross multiply
x³(6 + y) = 1 + 4y
6x³ + x³y = 1 + 4y

Step 5: Collect y terms
x³y – 4y = 1 – 6x³
y(x³ – 4) = 1 – 6x³
y = (1 – 6x³)/(x³ – 4)

Step 6: Simplify by factoring -1
y = -(6x³ – 1)/-(4 – x³)
y = (6x³ – 1)/(4 – x³)
Therefore: g⁻¹(x) = (6x³ – 1)/(4 – x³)