what is the inverse of the quadratic function f(x)=x^2+12x+37
f(x)=x^2+12x+37
y = x^2 + 12x + 37
Solve for x
x^2 + 12x = y - 37
Complete the square
x^2 + 12x + 36 = y - 37 + 36
(x + 6)^2 = y - 1
Take the square root of both sides
+- (x + 6) = (sqrt(y - 1))
x + 6 = (sqrt(y - 1))
-x - 6 = (sqrt(y - 1))
x = -6 +- (sqrt(y - 1))
Interchange x and y
y = -6 +- (sqrt(x - 1))
f^-1(x) = -6 +- (sqrt(x - 1))
To find the inverse of the quadratic function f(x) = x^2 + 12x + 37, follow these steps:
Step 1: Replace f(x) with y: y = x^2 + 12x + 37.
Step 2: Swap x and y: x = y^2 + 12y + 37.
Step 3: Solve the equation for y in terms of x. Rearrange the equation:
x = y^2 + 12y + 37
x - 37 = y^2 + 12y
Move all terms to one side:
y^2 + 12y - (x - 37) = 0.
Step 4: Solve the quadratic equation for y using the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / 2a,
where a = 1, b = 12, and c = -(x - 37).
Substituting these values, we get:
y = (-12 ± √(12^2 - 4(1)(-(x - 37)))) / (2(1))
Simplifying further:
y = (-12 ± √(144 + 4(x - 37))) / 2
y = (-12 ± √(144 + 4x - 148)) / 2
y = (-12 ± √(4x - 4)) / 2
y = (-12 ± 2√(x - 1)) / 2
Step 5: Simplify the expression:
y = (-12 ± 2√(x - 1)) / 2
Divide both numerator and denominator by 2:
y = -6 ± √(x - 1).
So, the inverse of the quadratic function f(x) = x^2 + 12x + 37 is given by:
f^(-1)(x) = -6 ± √(x - 1).
To find the inverse of a quadratic function, you can follow these steps:
Step 1: Substitute the function notation f(x) with y.
y = x^2 + 12x + 37
Step 2: Swap the x and y variables.
x = y^2 + 12y + 37
Step 3: Rearrange the equation to solve for y.
x - 37 = y^2 + 12y
Step 4: Complete the square on the right side of the equation.
x - 37 + 36 = y^2 + 12y + 36
x - 1 = (y + 6)^2
Step 5: Take the square root of both sides.
√(x - 1) = y + 6
Step 6: Solve for y by subtracting 6 from both sides.
y = √(x - 1) - 6
Therefore, the inverse of the quadratic function f(x) = x^2 + 12x + 37 is f^(-1)(x) = √(x - 1) - 6.