If f(x+(1)/(x))=x^2+(1)/(x^2)=g(x-(1)/(x)), then (a) f(1)=2=g(1) (b) f^'(1)=2=g^'(1) (c) (f ∘g)^

If f(x+(1)/(x))=x^2+(1)/(x^2)=g(x-(1)/(x)), then (a)...

If $f\left(x+\frac{1}{x}\right)=x^{2}+\frac{1}{x^{2}}=g\left(x-\frac{1}{x}\right)$, then (a) $\mathrm{f}(1)=2=\mathrm{g}(1)$ (b) $\mathrm{f}^{\prime}(1)=2=\mathrm{g}^{\prime}(1)$ (c) $(\mathrm{f} \circ \mathrm{g})^{\prime}(1)=4=\left(\mathrm{g} \mathrm{f}^{\prime}\right)(-1)$ (d) $(\mathrm{g} \circ \mathrm{f})^{\prime}(1)=-4=\left(\mathrm{f}^{\prime} \circ \mathrm{g}^{\prime}\right)(-1)$

If $f\left(x+\frac{1}{x}\right)=x^{2}+\frac{1}{x^{2}}=g\left(x-\frac{1}{x}\right)$, then
(a) $\mathrm{f}(1)=2=\mathrm{g}(1)$
(b) $\mathrm{f}^{\prime}(1)=2=\mathrm{g}^{\prime}(1)$
(c) $(\mathrm{f} \circ \mathrm{g})^{\prime}(1)=4=\left(\mathrm{g} \mathrm{f}^{\prime}\right)(-1)$
(d) $(\mathrm{g} \circ \mathrm{f})^{\prime}(1)=-4=\left(\mathrm{f}^{\prime} \circ \mathrm{g}^{\prime}\right)(-1)$

Key Concepts

Transformation of Variables

Transformation of variables involves substituting expressions (such as x + 1/x or x - 1/x) into a function to express it in a new form. This process allows one to exploit symmetries or simplify calculations by redefining the domain of the function. Such transformations are frequently employed in problems where functions are implicitly defined through these new variables.

Functional Equations and Implicit Differentiation

Functional equations are relationships where functions are defined implicitly by their values at transformed arguments, as seen with functions f and g in this problem. Implicit differentiation becomes necessary when the functions are not given in an explicit form, enabling the computation of derivatives by differentiating both sides of the defining equation with respect to the variable of interest.

Function Composition

Function composition refers to the process of applying one function to the result of another, forming a new function. It is fundamental in analyzing how layered processes transform inputs through successive operations. In this context, composing functions (such as evaluating (f ? g)(x) or (g ? f)(x)) requires understanding how the inner function’s output serves as the outer function’s input.

Chain Rule

The chain rule is a critical method in calculus used to differentiate composite functions. It states that the derivative of a composed function is the product of the derivative of the outer function (evaluated at the inner function) and the derivative of the inner function. This technique is essential when deciphering how small changes propagate through functions that have been redefined by transformations.

Transcript

00:01 Solving the problem f in x plus 1 over x equivalent to x squared as 1 over x squared equal to g of x minus 1 over x will be using the completing the square property to evaluate f in x and g in x.

00:14 So let us first use the property of complete the square.

00:18 So we'll obtain the value of fx plus 1 over x by complete the square property has x plus 1 over x whole squared minus 2.

00:32 Which implies that f in x is equivalent to x square minus 2 and similarly we'll obtain the value of g in x minus 1 over x by complete the square property as x minus 1 over x whole squared plus 2 this implies that the value of g in x is equivalent to x plus 2.

00:52 Now let us evaluate the value.

00:54 What are the f dash x which is equivalent to twice of x which is equivalent to g dash x which is equivalent to g dash x.

01:04 So, you can say that by looking at the functions, we can say that the value of f -dash -x is equal to g -dash -x for all the values of x.

01:15 Let us first evaluate the, let us substitute x to be equal to 1 in f -in -x.

01:21 So we're opting the f -in -1 to be equivalent to 1 -2, which is equal to negative of 1, and g -in -1 to be equivalent to 1 -2, which is equivalent to 3.

01:34 As we can see clearly that f in 1 is not equivalent to j in 1.

01:39 So we can see that the option a is not correct.

01:43 And as we can see that if an x is if dash x is equivalent to g dash x for all the values of x.

01:49 So we can see that f dash 1 is equivalent to g or 2 is equivalent to g dash 1...

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