Let be given by

MATHEMATICS · Multi-part · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

Let

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

be given by

f(x)=\dfrac{1}{x}

, where the domain is

S=\{x\in\mathbb{R}:|x|\ge1\}.

Answer the following questions. (i) Show that for every

x\in S

, the corresponding

y=f(x)

satisfies

|y|\le1

and

y\neq0

. (ii) Determine the exact range of

f

when the domain is

S

, writing your answer using interval notation. (iii) Verify explicitly that

y=1

and

y=-1

belong to this range, and explain why no value of

y

with

|y|>1

can occur. (iv) Sketch a rough graph indicating the original hyperbola

y=\dfrac{1}{x}

, the restricted domain

S

, and the corresponding range obtained in part (ii).