A function is described graphically as follows

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

A function

r

is described graphically as follows. • For

-3\le x\le 3

, the graph is the upper semicircle of radius

3

centred at the origin, that is,

y=\sqrt{9-x^2}

, and both endpoints are included. • For

4\le x<7

, the graph is the horizontal line

y=-2

. The point at

x=4

is an open circle, while the point at

x=7

is a solid dot. No other points belong to the graph. A new function

s

is defined by

s(x)=-r(x)+5

for every

x

in the domain of

r

. Answer the questions below. (i) Sketch the graph of

r

, indicating all open and closed points. (ii) Determine the domain and range of

r

. (iii) Sketch the graph of

s

on the same set of axes. (iv) Determine the domain and range of

s

, explaining clearly how they are obtained from those of

r

. State whether any endpoint changes from closed to open (or vice-versa).