Calculus: Solving Inverse Functions for f(x)=2x^3+3x^2+7x+4

• ace123
In summary, the conversation is about finding the inverse of a cubic function and using the inverse to solve a specific problem. The cubic formula is mentioned as a way to find the inverse, but it is deemed too complicated. Instead, the suggestion is to use the formula f^(-1)(f(x))=x, differentiate both sides, and use the chain rule to find the inverse. This method is considered easier than finding the general inverse.
ace123
Hi I'm trying to remember inverse functions for calculus but I'm having a few problems. So any help would be appreciated.

f(x)= 2x^3 + 3x^2 + 7x+ 4

So I have no clue how to solve this for the inverse. I know how to do basic ones. But I've forgotten these kind. So can i just get a step in the right direction.

You can get an inverse expression for a general cubic only by using the cubic formula. I DON'T recommend this. It's too complicated. What's the actual problem you need to solve? You probably don't need an explicit formula.

I'd agree, it would depend on the problem. You could use the cubic formula, but its pretty nasty.

Well I was hoping their was a nice way of doing b/c I knew about the cubic formula but wasn't about to use it for this. The actual question though was to find (f^-1)'(a) and the a= 4. I thought I had to find the inverse of the f(x) to solve the problem. Is there another way?

Yes there is. f^(-1)(f(x))=x. Differentiate both sides and use the chain rule. f^(-1)'(f(x))*f'(x)=1. So f^(-1)'(f(x))=1/f'(x). If you want to use this at f(x)=a=4, You still have to find a value of x such that f(x)=4. But that's a lot easier problem than finding the general inverse.

Oh, okay I see what your saying I didn't think about using f(x)=a=4. Thank you

What is a function and how is it related to calculus?

A function is a mathematical rule that assigns each input value to a unique output value. In calculus, functions are used to model and analyze changing quantities, such as the rate of change of a moving object or the slope of a curve.

What is an inverse function and why is it important in calculus?

An inverse function is a function that "undoes" the action of another function. In other words, if f(x) is a function, then its inverse function, denoted as f-1(x), will take the output of f(x) and return the original input. In calculus, inverse functions are important in solving equations and finding the area under a curve.

How do you find the inverse of a given function?

To find the inverse of a function, you can use the following steps:
1. Replace f(x) with y.
2. Switch the x and y variables, so the equation becomes x = 2y^3 + 3y^2 + 7y + 4.
3. Solve for y in terms of x. This will be the inverse function, f-1(x).
4. Verify the inverse by plugging in values for x and y, and checking if they satisfy both equations.

What is the domain and range of the given function and its inverse?

The domain of a function refers to all possible input values, while the range refers to all possible output values. For the given function f(x) = 2x^3 + 3x^2 + 7x + 4, the domain and range are all real numbers (-∞, ∞). The domain and range of its inverse, f-1(x), are also all real numbers (-∞, ∞).

How does solving inverse functions help in real-life applications?

Solving inverse functions is essential in many real-life applications, such as in finance, physics, and engineering. For example, in finance, inverse functions can be used to calculate compound interest rates or analyze stock market trends. In physics, inverse functions are used to model the motion of objects and calculate their velocities. In engineering, inverse functions are used to optimize designs and solve complex problems.

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