Math 100 - Studio College Algebra

Kansas State University

October 12, 2016

Math 100

Functions we’ve seen so far.

f (x) = c (constant),

f (x) = mx + b (linear),

f (x) = ax2 + bx + c (quadratic),

f (x) = |x | (absolute value),

f (x) =√x ,

Piecewise linear functions,

Compositions of these functions,

Other weird functions Brian makes up.

Math 100

Exponential Functions

Let a be a positive real number.

We will try to understand a function f (x) = ax .

We understand this function when x is an integer (i.e.x = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .).

Example: Consider 2x where x is an integer.

But we want this function to work for all real numbers x!

Math 100

How Do Exponential Functions Even Make Sense?

Recall that we can take a positive, real number a and take its n-throot where n is a positive integer ({1, 2, 3, 4, 5, . . .}):

n√a = a

1n .

Here, remember that n√a is the real number b (positive when n is

even) so thatbn = b · b · · · · · b︸ ︷︷ ︸

n times

= a.

A rational number is a real number that can be written as mn

where m and n are integers (or as a ratio of integers).Then we can define

amn =

(n√a)m

.

Well, at least ax makes sense when x is a rational number on thereal line!

Math 100

What about the “rest” of the real line?

Unfortunately, the precise answer to this question is technical. Buthere is an overview of the big picture:

It turns out that every real number can be “approximated” byrational numbers.

These “approximations” allow us to “extend” our function ax tothe irrational numbers by taking x to be a rational numbers whichis “closer and closer” to the irrational number we areapproximating.

Math 100

Approximation of 2√

2

x 2x

1 275 2.639015821545788518 . . .

141100 2.657371628193023161 . . .707500 2.664749650184043542 . . .70715000 2.665119088532351469 . . .

1414213100000 2.665143103797717989 . . .282847200000 2.665144027466092141 . . .√

2 2.665144142690225188 . . .

Math 100

Graph of y = 2x .

Math 100

Graph of y = 2x .

Growth as x →∞. As x →∞, 2x →∞.

Decay as x → −∞. As x → −∞, 2x → 0

Horizontal Asymptote at y = 0 as x → −∞.

Math 100

Graph of y = 2−x .

Math 100

Graph of y = 2−x .

Decay as x →∞. As x →∞, 2−x → 0.

Growth as x → −∞. As x → −∞, 2−x →∞.

Horizontal Asymptote at y = 0 as x →∞.

Math 100

Graph of y = 2x and graph of y = 2−x .

Related to one another by reflection about the y -axis.

Math 100

Graphs of y = 5x , 4x , 3x , 2x

Math 100

Example 1

Let

f (x) = 42x − 1.

1. Find f(12

).

2. Where does f (x) have a horizontal asymptote?

3. When does f (x) approach this asymptote?

Math 100

Euler’s Number e

e ≈2.71828182845904523536028747135266249775724709369995 . . .

Discovered by Jacob Bernoulli studying continuouslycompounded interest.

Leonhard Euler denoted it e and found several descriptions ofit, so his conventions stuck.

The function ex is called the exponential function.

Math 100

Graph of y = ex

Math 100

Logarithmic Functions

Since f (x) = ax is a one-to-one function, it has an inverse!!!

Math 100

Logarithmic Functions

Since f (x) = ax is a one-to-one function, it has an inverse!!!

Math 100

Logarithmic Functions

Since f (x) = ax is a one-to-one function, it has an inverse!!!

Math 100

Logarithmic Functions

From the picture, what is the domain of this inverse?

Math 100

Logarithmic Functions

Let a be a positive real number. Then the function loga(x) asksthe question “a raised to what exponent is equal to x?”We say that loga(x) is “log base a of x”. In other words, a iscalled the base of the log.

For any positive number a, ax and loga(x) are inverses of oneanother

log10(x) is called the common log

loge(x) = ln(x) is called the natural log

The domain of loga(x) is x > 0. The range (set of outputs) ofloga(x) is all real numbers.

Math 100

Example 2

Find log10(1000).

Math 100

Example 3

Find the domain of

log49208427482547π(x2 − 3x + 2)

Math 100