The functions f and g are inverse functions. The function f(x)=2 x^3+x-3 has an inverse function

step 1 Yeah, so we have this one as well. It's going to follow the same format as the one that we did i

The functions f and g are inverse functions. The function...

Key Concepts

Derivative of an Inverse Function

The derivative of an inverse function is determined by the relationship g'(y) = 1/f'(g(y)), where f is the original function and g is its inverse. This formula is derived from differentiating the identity f(g(y)) = y with respect to y, typically using the chain rule. It provides a useful method for calculating the slope of the inverse by evaluating the derivative of the original function at the corresponding point.

Inverse Functions

An inverse function reverses the mapping of an original function, meaning that if f produces an output from an input, the inverse function g returns the input from that output. The fundamental property of inverse functions is that f(g(x)) = x and g(f(x)) = x, which ensures that each function undoes the effect of the other.

Transcript

00:03 Yeah, so we have this one as well.

00:06 It's going to follow the same format as the one that we did in the last tutorial.

00:13 So we're going to find the g prime of 3 and g prime of 0.

00:20 First of all, we have the function given in the question.

00:24 You can see that f of 0 is negative 3.

00:27 So what is f of 0? f of 0 is negative 3.

00:33 And the f of 1 is 0, right, f of positive 1 is 0.

00:42 So this is what you have, after you put 0 and 1 in the function in the question, what is the derivative of the function? the derivative of the function right here is, you know, 6x squared plus 1.

00:57 6x squared plus 1.

00:59 So, you know, we've been told that g is an inverse of x.

01:03 Of f i beg important so if g is an inverse of f then if f of zero is given as negative three then g of negative three is going to give us back zero that is how inverse works right the function acts on one element and produce another element the inverse works backwards is going to work on this element and give us back the original one okay so if f of 0 is negative 3, then the inverse is going to act on the negative 3 and take us back to 0.

01:40 That is why you have this one because g is an inverse of x.

01:44 Of f.

01:45 What do i keep saying? g is an inverse of f.

01:50 And so if f of 1 is 0, then g of 0 is going to give us back the 1.

01:58 Okay.

02:00 So you have this one as the inverses.

02:03 Now you want to find g3...