Mathematics

Problem

1. (a) Define an exponential function and distinguish between the terms "exponential growth" and "exponential decay" in the context of its graph. (b) For the functions

f(x)=1.5^{x}

and

g(x)=\left(\dfrac1{1.5}\right)^{x}

, state in each case: (i) whether the curve is increasing or decreasing, (ii) the horizontal asymptote, (iii)

\displaystyle\lim_{x\to\infty}f(x)

and

\displaystyle\lim_{x\to\infty}g(x)

. Give reasons for each answer. (c) Explain why the point

(0,1)

always lies on the graph of

y=a^{x}

for any valid base

a

. (d) Describe, without calculation, how the graphs of

f

and

g

are related to each other geometrically.

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