what is the inverse of f(x) = 5(x-6)^2 ?

Question

what is the domain and range for this equation?

Answer 1

Given f(x) = 5(x-6)^2

y = 5(x-6)^2

5(x^2+36-12x) = y

y = 5x^2-60x+180

Inter change x and y.

To find inverse solve for y.

x = 5y^2-60y+180

Subtract x from each side.

5y^2-60y+(180-x) = 0

Compare it to quadratic equation ax^2+bx+c = 0.

a = 5, b = -60, c = 180-x

y = [60±√(3600-20(180-x)]/10

y = [60±√(3600-3600+20x)]/10

y = [60±√20x]/10

y = (60±2√5x)/10

y = 2(30±√5x)/10

y = (30±√5x)/5

y = 6±√5x/5

y = 6±√(x/5)

We know that all possible x values of the quadratic function are domain and all possible y values of quadratic function are range.

Domain is x ≥ 5 and range is y ≥ 0.

Comments

Inverse of 5(x - 6)^2 is 6 ± √(x/5).

Domain of 5(x - 6)^2 is all real numbers.