Question Video: Finding the Value of the Inverse Function of a Given Function Mathematics • Higher Education

Let 𝑓(π‘₯) = (1/2)π‘₯Β³ + (1/2)π‘₯Β² + 5π‘₯ βˆ’ 4 and let 𝑔 be the inverse of 𝑓. Given that 𝑓(2) = 12, what is 𝑔′(12)?

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Video Transcript

Let 𝑓 of π‘₯ be equal to one-half π‘₯ cubed plus one-half π‘₯ squared plus five π‘₯ minus four and let 𝑔 be the inverse of 𝑓. Given that 𝑓 of two is equal to 12, what is 𝑔 prime of 12?

In order to help us find 𝑔 prime of 12, we can use the formula for derivatives of inverse functions. This tells us that if 𝑔 is the inverse function of 𝑓, then 𝑔 prime of 𝑦 is equal to one over 𝑓 prime of 𝑔 of 𝑦. Let’s start by finding 𝑓 prime of π‘₯, the derivative of 𝑓 with respect to π‘₯. We can see that 𝑓 is a polynomial. Therefore, in order to find its derivative, we can differentiate it term by term using the power rule for differentiation. We simply multiply by the power and decrease the power by one. This gives us that 𝑓 prime of π‘₯ is equal to three over two π‘₯ squared plus π‘₯ plus five.

Next, we’ll observe the fact that we’re trying to find 𝑔 prime of 12. And so, we can substitute 𝑦 equals 12 into our formula for 𝑔 prime of 𝑦. This gives us that 𝑔 prime of 12 is equal to one over 𝑓 prime of 𝑔 of 12. Now, we do not know what 𝑔 of 12 is. However, we have been given in the question that 𝑓 of two is equal to 12. And since 𝑔 is the inverse function of 𝑓, we can apply 𝑔 to both sides here. And we’ll obtain that 𝑔 of 12 is equal to two. This is because of the way inverse functions work. If we take 𝑔 of 𝑓 of two, then we’ll simply get two.

We can now substitute this value of 𝑔 of 12 back into our equation for 𝑔 prime of 12. And we obtain that it’s equal to one over 𝑓 prime of two. Now, we’ve already found 𝑓 prime of π‘₯. So we can simply substitute π‘₯ equals two in order to find 𝑓 prime of two. And we obtain that 𝑓 prime of two is equal to 13. And substituting the value of 𝑓 prime of two back into 𝑔 prime of 12, we obtain that 𝑔 prime of 12 is equal to one over 13.

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