Math Problem Statement
The function ff is defined by f, of, x, equals, minus, x, cubed, minus, 3, x, minus, 5f(x)=−x3−3x−5 and the point left parenthesis, 1, comma, minus, 9, right parenthesis(1,−9) is on the graph of f, .f. If g, of, x, equals, f, to the power minus 1 , left parenthesis, x, right parenthesisg(x)=f−1(x), what is the value of g, prime, of, minus, 9, question markg′(−9)?
Answer
Solution
To find the value of , where , we can use the relationship between the derivatives of inverse functions. Specifically, if , then:
In this case, we're interested in . Notice that since the point is on the graph of , it means that:
So, . To find , we need to calculate first.
Given , the derivative is:
Now, substitute into :
Therefore:
So, the value of is .
Would you like any further details or have any other questions?
Here are 5 related questions you might find interesting:
- How do you find the inverse of a cubic function algebraically?
- What are the conditions for a function to have an inverse?
- Can a function with a negative derivative have an increasing inverse?
- How would you find the derivative of an inverse function without knowing the original function explicitly?
- How does the graph of a function compare to the graph of its inverse?
Tip: When dealing with derivatives of inverse functions, always remember the key relationship and verify that the function is one-to-one.
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Derivatives
Formulas
Derivative of a cubic function
Derivative of an inverse function
Theorems
Inverse function theorem
Suitable Grade Level
Advanced High School
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