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Math 1051 Precalculus I Lecture Notes
5.1 Composite Functions
Given AND
find (f ο g)(x)
Domain f:  Domain g:  Domain 
x ≠ 0  
So  x ≠ 0  x ≠ 2 
Domain of (f ο g)(x) must respect domains of g and. So, we get
{x  x ≠ 0, x ≠ 2}
Domain of a composite function
5.2 OnetoOne Functions & Inverse Functions
Determine whether a function is onetoone
Use horizontal line test on the graph of a given function.
Inverse of a function by mapping
Graph of the inverse of a function
The function and its inverse are symmetric about the line y = x.
Inverse of a function given its equation
pg 261, #68: Find the inverse of
So, the inverse function is
Domain of f:
{x  x ≠ 2}
Range of f = Domain of f inverse:
{y  y ≠ 3}
Here is the graph. Note the symmetry of the function and its inverse about the
line y = x .
5.3 Exponential Functions
Definition
f (x) = a^{x} a > 0, a ≠ 1
Graph exponential functions
Define the number e
Solve exponential equations
pg 274 #70: Solve
We can write the exponential functions with the same base and then equate the
exponents:
These are both in the domain of the original equation.
Or, if we could not write the expressions with the same bases, we would use
logs:
Now, we can use the quadratic formula to solve this:
So, we get the same answer as before.
5.4 Logarithmic Functions
Change between log and exponential forms
Evaluate logarithmic expressions
Domain
Graph logarithmic functions
5.5 Properties of Logarithms
Properties
Memorize the properties
Function operating on its inverse:
Function operating on its inverse:
Log of a product :
log_{a}MN = log_{a}M + log_{a}N
Log of a quotient:
Log of a power :
Equal exponentials: If a^{M}
= a^{N}
then M = N ,
M, N, and a are positive real numbers , a ≠ 1
Equal logs:
If log_{a}M = log_{a}N then M = N ,
M, N, and a are positive real numbers, a ≠ 1
Rewrite logarithmic expressions using properties
Changeofbase
formula
5.6 Logarithmic and Exponential Equations
Solve logarithmic equations
pg 303 #24: Solve
log_{5}(x + 3) = 1 log_{5}(x 1).
Put all logs together on one side of equation and then change to exponential:
4 is not in the domain of the original equation so the only solution is x = 2.
Solve exponential equations
Solve: 81^{x}
+ 2 = 3 • 9^{x}
Can we make the bases of the exponentials the same?
Now, we have 9^{x} and so try a u substitution :
Now, back substitute:
5.7 Compound Interest
Determine future value of a lump sum of money
Calculate effective rates of return
Determine present value of a lump sum of money
Memorize the formulas.
Simple interest : I = Prt
Compound interest:
Continuous compounding: A = Pe^{rt}
Effective rate of interest:
I_{simple}= I_{compound}
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