The function is defined by

MATHEMATICS · Subjective · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

5. The function

g

is defined by

g(x)=3^{-(x-2)}+4

. (a) Express

g(x)

as a sequence of transformations of the base curve

y=3^{x}

, stating clearly the order and nature of each transformation. (b) Find the equation of the horizontal asymptote of the graph of

g

. (c) Determine the coordinates of the

y

-intercept and the

x

-intercept (if any exist). (d) Describe the end behaviour of the graph, that is, evaluate

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

and

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

. (e) Sketch the graph of

y=g(x)

, clearly showing the asymptote, intercepts, and the point corresponding to

x=2