# 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.

Graph of the inverse of a function

**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
Change-of-base
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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