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Math 100 - Studio College Algebra
Kansas State University
October 12, 2016
Math 100
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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
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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
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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
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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
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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
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Graph of y = 2x .
Math 100
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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
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Graph of y = 2−x .
Math 100
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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
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Graph of y = 2x and graph of y = 2−x .
Related to one another by reflection about the y -axis.
Math 100
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Graphs of y = 5x , 4x , 3x , 2x
Math 100
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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
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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
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Graph of y = ex
Math 100
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Logarithmic Functions
Since f (x) = ax is a one-to-one function, it has an inverse!!!
Math 100
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Logarithmic Functions
Since f (x) = ax is a one-to-one function, it has an inverse!!!
Math 100
![Page 18: Math 100 - Studio College Algebrarekha/SCAFall2018/Lecture08.pdf · a is the real number b (positive when n is even) so that bn = b| b {z b} n times = a: A rational number is a real](https://reader034.vdocuments.mx/reader034/viewer/2022042711/5f6ff20a1fdfde08b537c2e2/html5/thumbnails/18.jpg)
Logarithmic Functions
Since f (x) = ax is a one-to-one function, it has an inverse!!!
Math 100
![Page 19: Math 100 - Studio College Algebrarekha/SCAFall2018/Lecture08.pdf · a is the real number b (positive when n is even) so that bn = b| b {z b} n times = a: A rational number is a real](https://reader034.vdocuments.mx/reader034/viewer/2022042711/5f6ff20a1fdfde08b537c2e2/html5/thumbnails/19.jpg)
Logarithmic Functions
From the picture, what is the domain of this inverse?
Math 100
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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
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Example 2
Find log10(1000).
Math 100
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Example 3
Find the domain of
log49208427482547π(x2 − 3x + 2)
Math 100