 # 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 One-to-One Functions & Inverse Functions

Determine whether a function is one-to-one
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) = ax 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 : logaMN = logaM + logaN
Log of a quotient: Log of a power : Equal exponentials: If aM = aN then M = N ,
M, N, and a are positive real numbers, a ≠ 1

Equal logs: If logaM = logaN then M = N ,
M, N, and a are positive real numbers , a ≠ 1

Rewrite logarithmic expressions using properties
Change-of-base formula  ## 5.6 Logarithmic and Exponential Equations Solve logarithmic equations
pg 303 #24: Solve log5(x + 3) = 1- log5(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: 81x + 2 = 3 • 9x
Can we make the bases of the exponentials the same? Now, we have 9x 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 = Pert
Effective rate of interest:
Isimple= Icompound Prev Next