Mathematics

Problem

Let

f,\,g:\mathbb{R}\to\mathbb{R}

be defined by

f(x)=2x+3

and

g(x)=x^{2}

. Let

I_{\mathbb{R}}

denote the identity function on

\mathbb{R}

. a. (i) Find

(f\circ I_{\mathbb{R}})(x)

and

(I_{\mathbb{R}}\circ f)(x)

. Hence verify that both expressions are equal to

f(x)

. b. (ii) Compute

(g\circ I_{\mathbb{R}})(5)

and

(I_{\mathbb{R}}\circ g)(5)

. c. (iii) Determine

(f\circ g)(x)

and

(g\circ f)(x)

. Are the two compositions equal? Give a reason for your answer. d. (iv) Using the results above, explain the special role played by the identity function in compositions compared with an arbitrary function.

Browse the problem library