If f is a one-to-one function with f(3)=8 and f′(3)=7, find th...

Question

If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.

Accepted Answer

Welcome back, everyone. Determine the equation of the line tangent to Y equals J inverse of X at X equals 5. If J 2 equals 5 and G 2 equals -2 were given for answer choices A Y equals -12 x minus 9/2 B Y equals -12 X plus 9/12s. C Y equals 1/2. X plus 9/12s and D Y equals -1/2 x plus 5 halves. So let's recall the equation of the tangent line. It is y equals f the derivative of a function at the point of interest X not multiplied by X minus X, and then we're adding the value of the function at that point of tangency. So in this case, well, we're going to replace F with J, right? So we're going to get J. Inverse, right, because this is the function of interest, we want to find the equation of the line tangent to jinverse. So we're going to take J inverse at point X not multiplied by x minus X plus the value of the function at that point. So we take J inverse at point X0. So what do we know? Well, first of all, we know the point of tangency. We know that X is equal to 5 based on the context of the problem. Now what we want to do is evaluate G inverse at 0.5, and what we're going to do is simply apply one of the properties of derivatives. We have to recall that this is equal to 1 divided by. The derivative of J, the parent function at a point of J inverse of 5. Now, if we carefully analyze the context, what do we know? Well, essentially it says that J of 2 equals 5, right? So we know that J of 2. Equals 5, and we have to recall that whenever we consider the relationship between the parent and the inverse function, well, we are simply swapping the domain and a range. So this means that if we take j inverse of 5, this is going to be equal to 2. So essentially we can continue this equation and we can say that this is equal to 1 divided by J at 0.2. Which is given to us, right? We know that this is equal to. -2. So in the denominator, we simply have -2, and if we divide 1 by 2, we're going to get -12. So this is the first part that we want to identify. We want to identify the slope of the tangent line. It is equal to -12, and then we want to evaluate J inverse at X not. Now, we already know that it is equal to 2 based on the previous part. So what we're going to do is use the equation of the tangent line and we're going to substitute the values. So we're going to get -12 multiplied by X minus the point of tangency, so X minus 5 plus the value of the function at 5, which is 2. And now let's distribute the terms. That's -1/2 x plus because we're multiplying two negative numbers, so plus 5 halves plus 2, which can be rewritten as 4 divided by 2, and this means that Y equals -12 x plus 9 divided by 2. Which corresponds to the answer choice B, and that's our final answer. Thank you for watching.

If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.

Key Concepts

Inverse Function

Derivative of Inverse Functions

Tangent Line Equation

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