Chapter Nineteen — Exponential Functions

The Humongous Book of Algebra Problems

418

Graphing Exponential Functions

Graphs that start close to y = 0 and climb fast

Note: Problems 19.1–19.5 refer to the function j(x) = 2

x

.

19.1 Complete the table of values below.

Evaluate j(x) for each of the x-values in the left column.

Note: Problems 19.1–19.5 refer to the function j(x) = 2

x

.

19.2 Graph j(x) using the table of values generated in Problem 19.1.

According to Problem 19.1, the graph of j(x) passes through the following

points: , , (0,1), (1,2), (2,4), and (3,8). The graph of j(x) is

presented in Figure 19-1.

Chapter Nineteen — Exponential Functions

The Humongous Book of Algebra Problems

419

Figure 19-1: The graph of j(x) = 2

x

.

Note: Problems 19.1–19.5 refer to the function j(x) = 2

x

.

19.3 Identify the domain, range, and intercept(s) of j(x).

Consider the graph of j(x) in Figure 19-1. Every vertical line drawn on the

coordinate plane that intersects the graph represents a member of the domain

of j(x). Any vertical line drawn on the graph will intersect j(x), so the domain of

j(x) is all real numbers.

Every horizontal line that intersects the graph of j(x) represents a member

of the range of the function. The line y = 0, the x-axis, does not intersect the

graph, and neither does any horizontal line below it. Only horizontal lines

above y = 0 intersect the graph, so the range of j(x) is j(x) > 0.

The graph of j(x) intersects the y-axis at point (0,1), because j(0) = 2

0

= 1.

Therefore, the y-intercept of j(x) is 1.

Note: Problems 19.1–19.5 refer to the function j(x) = 2

x

.

19.4 Demonstrate that f(x) = log

2

x (from Problems 18.11–18.12) and j(x) are

inverses based on their graphs.

According to Problem 18.11, the graph of f(x) = log

2

x passes through points

, , (1,0), (2,1), (4,2), and (8,3). According to Problem 19.1, the

graph of j(x) = 2

x

passes through points , , (0,1), (1,2), (2,4), and

(3,8). Reversing the coordinate pairs of one graph produces the coordinate

pairs of the other, a characteristic of inverse functions. Furthermore, consider

the graphs of f(x) and j(x), plotted on the same coordinate plane in Figure 19-2.

The x-axis

is an asymptote

of the graph, so j(x)

gets innitely close to

the x-axis but never

touches it. That means

j(x) doesn’t have an

x-intercept.

Chapter Nineteen — Exponential Functions

The Humongous Book of Algebra Problems

420

Figure 19-2: The graphs of f(x) = log

2

x and j(x) = 2

x

are reﬂections of each other across

the dotted line y = x.

The graph of j(x) is the graph of f(x) reﬂected across the line y = x (and vice

versa), indicating that reversing all the points on one graph will generate the

points on the other.

Note: Problems 19.1–19.5 refer to the function j(x) = 2

x

.

19.5 Graph .

To transform the function j(x) = 2

x

into , multiply the input by

–1 to get –x (which reﬂects the graph of j(x) across the y-axis) and subtract 3

(which shifts the graph down three units). The graph of is presented in

Figure 19-3.

The table

of values for

f(x) and j(x)

prove that six

coordinate pairs

are reversed copies

of each other.

When functions are

reections of each

other (and the

dotted line

y = x in Figure 19-2

is the mirror), all

of the points on

the graphs are

reversed copies

of each

other.

Chapter Nineteen — Exponential Functions

The Humongous Book of Algebra Problems

421

Figure 19-3: The graph of is the graph of j(x) = 2

x

reﬂected across the

y-axis and shifted down three units.

Note: Problems 19.6–19.9 refer to the function g(x) = e

x

.

19.6 Use a calculator to complete the table of values below, rounding all values of

g(x) to the thousandths place.

Evaluate the natural exponential function e

x

for each value of x in the left

column.

Euler’s

number (e) is

the base for the

natural exponential

function e

x

, just like

it’s the base for the

natural logarithmic

function log

e

x = ln

x.

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