Question

For which pairs of functions is (f•g)(x) = x?

fx)=x^2 and g(x)= 1/x
f{x) = 2/x and g(x) = 2/x
f(x)= x-2/3 and g(x) = 2-3/x
f(x) = 1/2x - 2 and g(x) = 1/2x+ 2

Answer 1

 Rules for the operations of the functions state,

• (f o g)(x) = f[g(x)]

Option (2) is the correct option.

    Now we will apply this operation in each option to check (f o g)(x) = x

Option (1)

f(x) = x² and g(x) = $\frac{1}{x}$

Therefore, (f o g)(x) = f[g(x)]

                                = $\frac{1}{x^2}$

False.

Option (2)

f(x) = $\frac{2}{x}$ and g(x) =

(f o g)(x) = $\frac{2}{\frac{2}{x}}$

             = $\frac{2}{1}\times \frac{x}{2}$

             = x

True.

Option (3)

f(x) = $\frac{x-2}{3}$ and $g(x) = 2-3x$

(f o g)(x) = f[g(x)]

             = $\frac{2-3x-2}{3}$

             = -x

False.

Option (4)

f(x) = $\frac{1}{2x}-2$ and $g(x)=\frac{1}{2x}+2$

(f o g)(x) = f[g(x)

             = $\frac{1}{2(\frac{1}{2x}+2)}-2$

             = $\frac{1}{\frac{1}{x}+4}-2$

             = $\frac{x}{1+4x}-2$

             = $\frac{x-2(4x+1)}{4x+1}$

             = $\frac{x-8x-2}{4x+1}$

             = $-\frac{7x+2}{4x+1}$

False.

                Therefore, Option (2) is the correct option.

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