what is the inverse of the quadratic function f(x)=x2+12x+37
To find the inverse of a quadratic function, we need to follow a few steps:
Step 1: Replace f(x) with y:
y = x^2 + 12x + 37
Step 2: Swap the variables x and y:
x = y^2 + 12y + 37
Step 3: Solve for y:
Bring the constant term, 37, to the left side:
x - 37 = y^2 + 12y
Complete the square by dividing the coefficient of y by 2 and squaring it (6^2 = 36), then add this value to both sides of the equation:
x - 37 + 36 = y^2 + 12y + 36
x - 1 = (y + 6)^2
Step 4: Take the square root of both sides:
±sqrt(x - 1) = y + 6
Step 5: Isolate y:
y = -6 ± sqrt(x - 1)
Thus, the inverse of the quadratic function f(x) = x^2 + 12x + 37 is given by:
f^(-1)(x) = -6 ± sqrt(x - 1)