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Question Find the inverse of the quadratic function f(x)=x2+6x+15f(x) = x^2 + 6x + 15 by completing the square.

Studdy Solution

STEP 1

Assumptions
1. The given quadratic function is f(x)=x2+6x+15 f(x) = x^2 + 6x + 15 .
2. Completing the square involves rewriting the quadratic in the form (x+h)2+k (x + h)^2 + k where h h and k k are constants.
3. The inverse function, denoted as f−1(x) f^{-1}(x) , will be found after completing the square and will satisfy the condition f(f−1(x))=x f(f^{-1}(x)) = x .
4. The domain of the inverse function must be restricted to ensure it is a function.

STEP 2

First, we will complete the square for the quadratic function f(x)=x2+6x+15 f(x) = x^2 + 6x + 15 . To do this, we need to form a perfect square trinomial from the quadratic and linear terms.
f(x)=x2+6x+(62)2−(62)2+15 f(x) = x^2 + 6x + \left(\frac{6}{2}\right)^2 - \left(\frac{6}{2}\right)^2 + 15

STEP 3

Calculate the square of half the coefficient of x x and add and subtract this value inside the function.
f(x)=x2+6x+9−9+15 f(x) = x^2 + 6x + 9 - 9 + 15

STEP 4

Rewrite the quadratic function by grouping the perfect square trinomial and combining the constants.
f(x)=(x2+6x+9)−9+15 f(x) = (x^2 + 6x + 9) - 9 + 15

STEP 5

Recognize the perfect square trinomial and rewrite it as a squared binomial.
f(x)=(x+3)2−9+15 f(x) = (x + 3)^2 - 9 + 15

STEP 6

Combine the constants to simplify the expression.
f(x)=(x+3)2+6 f(x) = (x + 3)^2 + 6

STEP 7

Now that we have completed the square, we can find the inverse function by replacing f(x) f(x) with y y and then solving for x x .
y=(x+3)2+6 y = (x + 3)^2 + 6

STEP 8

To find the inverse, we switch x x and y y .
x=(y+3)2+6 x = (y + 3)^2 + 6

STEP 9

Next, we solve for y y to get the inverse function. Start by isolating the squared term.
(y+3)2=x−6 (y + 3)^2 = x - 6

STEP 10

Take the square root of both sides, remembering to consider both the positive and negative roots.
y+3=±x−6 y + 3 = \pm\sqrt{x - 6}

STEP 11

Isolate y y to find the inverse function.
y=−3±x−6 y = -3 \pm\sqrt{x - 6}

STEP 12

Since we need the inverse to be a function, we must choose either the positive or negative root. The original function f(x) f(x) is an upward-opening parabola, so its inverse will be the top half of a sideways parabola. Therefore, we choose the positive root.
y=−3+x−6 y = -3 + \sqrt{x - 6}

STEP 13

Finally, write the inverse function in function notation.
f−1(x)=−3+x−6 f^{-1}(x) = -3 + \sqrt{x - 6}
The equation of the inverse function is f−1(x)=−3+x−6 f^{-1}(x) = -3 + \sqrt{x - 6} .

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