Let be defined by and let be defined by .

MATHEMATICS · Multi-part · GCSE · Mathematics · Algebra · Algebra

Problem

Let

p:\mathbb R\to\mathbb R

be defined by

p(x)=2x-5

and let

q:\mathbb R\to\mathbb R

be defined by

q(x)=\dfrac{x+5}{2}

. (a) Calculate

p\bigl(q(7)\bigr)

and

q\bigl(p(7)\bigr)

. (b) Without assuming any prior relationship between

p

and

q

, compute the general compositions

p\circ q

and

q\circ p

and simplify each expression. (c) From the results in part (b), determine whether

p

and

q

are inverses of each other. Give a brief justification referencing the defining property of inverse functions.