Two functions are defined by and .

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Two functions are defined by

p(x)=2^{x}

and

q(x)=5(0.5)^{x}

. (i) Find the

y

-intercepts of both functions. (ii) Evaluate

p(3)

q(3)

p(-2)

and

q(-2)

. (iii) As

x\to \infty

, determine which of the two curves approaches the

x

-axis more rapidly. Justify your answer using the properties of exponential functions. (iv) On the same set of axes, sketch the graphs of

p(x)

and

q(x)

, labelling all points calculated in parts (i) and (ii) as well as the common horizontal asymptote.