Studdy AI Logo
Math

QuestionFind the inverse of the function f(x)=−4x2f(x)=-4 x^{2} and express it as f−1(x)=f^{-1}(x)=.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=−4xf(x)=-4 x^{} . We are asked to find the inverse of the function f(x)f(x)

STEP 2

The first step in finding the inverse of a function is to replace f(x)f(x) with yy. This gives usy=−4x2y = -4x^{2}

STEP 3

The next step is to swap xx and yy. This gives usx=−y2x = -y^{2}

STEP 4

Now we need to solve this equation for yy to get the inverse function. To do this, we first divide both sides by −4-4x−4=y2\frac{x}{-4} = y^{2}

STEP 5

Next, we take the square root of both sides. However, we must remember that when we take the square root of a square, we get two solutions a positive and a negative. Therefore, we gety=−x4andy=−−x4y = \sqrt{\frac{-x}{4}} \quad \text{and} \quad y = -\sqrt{\frac{-x}{4}}

STEP 6

implify the square root by taking the square root of4 in the denominatory=−x2andy=−−x2y = \sqrt{\frac{-x}{2}} \quad \text{and} \quad y = -\sqrt{\frac{-x}{2}}

STEP 7

Now, we have found the inverse of the function f(x)=−4x2f(x)=-4 x^{2}. However, we must remember that the original function f(x)f(x) is a function, which means it passes the vertical line test. The inverse, however, does not pass the vertical line test, which means it is not a function. Therefore, we usually choose one branch of the inverse to be the function. In this case, we choose the positive branchf−1(x)=−x2f^{-1}(x) = \sqrt{\frac{-x}{2}}

STEP 8

To draw the inverse function, we can plot the points that satisfy the equation f−1(x)=−x2f^{-1}(x) = \sqrt{\frac{-x}{2}}. We can also reflect the graph of the original function f(x)=−4x2f(x)=-4 x^{2} over the line y=xy=x to get the graph of the inverse function.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord