Question

Show that (fof-1) (x)=x and (F-1 of)(x) = x for the following pair of functions.f(x) = 5-4x, f'(x) =5-754Show that (fof-1)(x) = x.(fof- ')(x) = 0Write the expression for the composition.= x(Do not simplify. Type an exact answer, using radicals as needed.)

Answer 1

Evacuating the function and it's inverse can be shown to be equal to x by substitution

How to show the substitution

The given data includes

f(x) = ⁵√(5 - 4x)

f⁻¹(x) = (5 - x⁵) / 4

Substituting for x in f/9x) we have

f(x) = ⁵√(5 - 4((5 - x⁵) / 4 )

`f(x) = \sqrt[5]{5-4(5-x^(5) )/(4) }`

`f(x) = \sqrt[5]{5-(5-x^(5)) } }`

`f(x) = \sqrt[5]{5-5+x^(5)}`

`f(x) = \sqrt[5]{x^(5)}`

f(x) = x

Hence we can say that (f ° f⁻¹) (x) = x

Answer 2

To obtain the expression for the composition of the function of the inverse function of x, the following steps are necessary:

Step 1: Write out the expression for the function of x and the inverse function of x, as below:

`\begin{gathered} f(x)=\sqrt[5]{5-4x} \\ ^{}f^(-1)(x)=(5-x^5)/(4) \end{gathered}`

Step 2: Write out the expression for the composition of the function of the inverse function of x, as below:

`(f^(-1)Of))(x)=^{}\frac{5-(\sqrt[5]{5-4x})^5}{4}`

Thus, the above is how the expression for the composition of the function of the inverse function of x is to be written out