# 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

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