f(x) = x2 + 11x, x ≥ − 11/2
find the inverse
y = x^2+11x
for inverse, switch names and solve for y:
x = y^2+11y
y^2+11y-x = 0
y = (-11±√(121+4x))/2
with the original domain restricted to x >= -11/2, that means
y = -11/2 + √(121+4x)/2
or, consider that
y = x^2+11x
= (x+11/2)^2 - 121/4
restricting the domain to x >= -11/2 means we take just the right branch of the parabola. So, swapping names,
x = (y+11/2)^2 - 121/4
y+11/2 = +√(x + 121/4)
and we get the above result
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To find the inverse of the function f(x) = x^2 + 11x, we can follow these steps:
Step 1: Replace the function notation f(x) with y.
y = x^2 + 11x
Step 2: Swap the positions of x and y. The equation becomes:
x = y^2 + 11y
Step 3: Solve the equation for y (the inverse function).
x = y^2 + 11y
Step 4: Rearrange the equation to isolate y:
0 = y^2 + 11y - x
Step 5: Now we have a quadratic equation, which can be solved using the quadratic formula:
y = (-11 ± √(11^2 - 4(1)(-x))) / (2(1))
Simplifying this expression further:
y = (-11 ± √(121 + 4x)) / 2
So, the inverse function of f(x) = x^2 + 11x is given by:
f^(-1)(x) = (-11 ± √(121 + 4x)) / 2.
To find the inverse of the given function f(x) = x^2 + 11x, we need to follow these steps:
Step 1: Replace f(x) with y: y = x^2 + 11x.
Step 2: Swap x and y: x = y^2 + 11y.
Step 3: Rearrange the equation to solve for y: x - y^2 = 11y.
Step 4: Group the y terms on one side: x = y^2 + 11y.
Step 5: Complete the square to get the quadratic equation in standard form: y^2 + 11y + (11/2)^2 = x + (11/2)^2.
Step 6: Simplify the right side: y^2 + 11y + (121/4) = x + (121/4).
Step 7: Factor the left side: (y + 11/2)^2 = x + (121/4).
Step 8: Take the square root of both sides: y + 11/2 = √(x + 121/4).
Step 9: Isolate y by subtracting 11/2 from both sides: y = -11/2 ± √(x + 121/4).
So, the inverse of the function f(x) = x^2 + 11x is given by: f^(-1)(x) = -11/2 ± √(x + 121/4).