Let be defined by and

MATHEMATICS · Multi-part · CBSE · Mathematics · Relations and Functions · Relations and Functions

Problem

Let f, g:R→Rf,\,g:\mathbb{R}\to\mathbb{R} be defined by f(x)=2x+3f(x)=2x+3 and g(x)=x2g(x)=x^{2}. Let IRI_{\mathbb{R}} denote the identity function on R\mathbb{R}. (i) Find (f∘IR)(x)(f\circ I_{\mathbb{R}})(x) and (IR∘f)(x)(I_{\mathbb{R}}\circ f)(x). Hence verify that both expressions are equal to f(x)f(x). (ii) Compute (g∘IR)(5)(g\circ I_{\mathbb{R}})(5) and (IR∘g)(5)(I_{\mathbb{R}}\circ g)(5). (iii) Determine (f∘g)(x)(f\circ g)(x) and (g∘f)(x)(g\circ f)(x). Are the two compositions equal? Give a reason for your answer. (iv) Using the results above, explain the special role played by the identity function in compositions compared with an arbitrary function.

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