Mathematics

Problem

Consider the three expressions

f(x)=3^{x}, \qquad g(x)=\left(\dfrac13\right)^{x}, \qquad h(x)=(-2)^{x}.

Answer the following questions. (i) Classify

f

and

g

as growth or decay functions and state whether

h

is an exponential function on

\mathbb R

. (ii) Prove that

f(x)\,g(x)=1

for every real

x

, and deduce that

f(x)>1>g(x)

whenever

x>0

. (iii) Use the result of part (ii) to establish the inequalities

g(x)<g(y)<f(y)<f(x)

whenever

x>y>0

. (iv) Discuss the monotonicity of

h(x)

when

x

is restricted to natural numbers, and explain why no such monotonicity statement can be extended to all real

x