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Exponential Functions: Definition, Graphs, Laws and Examples - Prof. Erin Mullen, Study notes of Mathematics

University of Connecticut (UConn) - Avery PointMathematics
Prof. Erin Mullen

The concept of exponential functions, including their mathematical representation, graphs, laws of exponents, and examples. It explains how to graph exponential functions, simplify expressions, and solve equations. It also introduces the number e.

Typology: Study notes

Pre 2010

Uploaded on 02/24/2010

koofers-user-5og
koofers-user-5og 🇺🇸

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Section 1.5 1
Section 1.5: Exponential Functions
An exponential function can be written in the form
y=f(x) = ax,
where ais a positive constant.
What does this mean? Consider the following cases:
•x=n, where nis a positive integer
•x= 0
•x=−n, where nis a positive integer
•x=p
q, where pand qare integers, q > 0(this is a rational number)
•xis an irrational number, i.e, not rational
Graphs of Exponential Functions
•a > 1
•0<a<1
pf3

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Download Exponential Functions: Definition, Graphs, Laws and Examples - Prof. Erin Mullen and more Study notes Mathematics in PDF only on Docsity!

Section 1.5: Exponential Functions

An exponential function can be written in the form

y = f (x) = ax,

where a is a positive constant. What does this mean? Consider the following cases:

  • x = n, where n is a positive integer
  • x = 0
  • x = −n, where n is a positive integer
  • x =

p q

, where p and q are integers, q > 0 (this is a rational number)

  • x is an irrational number, i.e, not rational

Graphs of Exponential Functions

  • a > 1
  • 0 < a < 1

Example 1. Graph the following exponential function using transformations and state the domain and range.

f (x) = 2 − 4 −x

Laws of Exponents If a and b are positive numbers and x and y are any real numbers,

  1. ax+y^ = axay.
  2. ax−y^ =

ax ay^

  1. (ax)y^ = axy^ = (ay)x.
  2. (ab)x^ = axbx.
  3. ax^ = ay^ ⇐⇒ x = y

Non-Laws of Exponents

  1. ax+y^6 = ax^ + ay.
  2. a(bx) 6 = (ab)x.
  3. (a + b)x^6 = ax^ + bx.

So f (x) = 2 − 4 −x^ = 2 −

)x and g(x) = 2x−^2 = 2x 2 −^2 =

2 x, but h(x) = 3(2x) 6 = 6x.

Example 2. Simplify the following expression so that it is written as a constant times a power of x. √ (^38) x

2 x

x

Example 3. Solve the equation 2 · 16 x^ = 8x−^1 for x.