General Mathematics

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General Mathematics by Mind Map: General Mathematics

1. Functions and Their Graphs

1.1. denoted by f, is rule assigns to each element x in the set X exactly one element f(x)

1.2. f is the dependent variable while (x) is the independent variable.

1.3. Piesewise-defined function is a function that is defined by two or more equations over a specified domain.

1.4. Absolute value |x|

1.5. Radical functions - functions with radical sign.

1.6. Vertical line test -  the should only intersect once in the vertical line for the equation to be considered as a function.

1.7. Relative maximum - highest point of the graph

1.8. Relative Minimum - lowest point of the graph

1.9. Asymptotes

1.9.1. vertical - right to left

1.9.2. horizontal- up or down

1.10. Rational function - it is an equation in fraction form and the denominator should not be equal to zero

1.11. Composition of the functions, sum, difference, product, and quotient.

1.12. Rational equation is an equation containing at least one rational expression.

1.13. Extraneous solution is a solution that does not solve the original equation.

1.14. Rational inequality is a mathematical statement that relates a rational expression as either less than or grater than another.

1.15. Zeroes of both numerator and the denominator of the rational expression are called critical numbers.

2. Exponential and Logarithmic Functions

2.1. Exponential

2.1.1. If f is increasing or decreasing throughout domain, then f is one-to-one

2.1.2. if each horizontal line intersects the graph in at most one point, then the function if one-to-one.

2.1.3. Laws of exponent

2.1.3.1. (a^x)(a^y)=a^x+y

2.1.3.2. (a^x)^Y=a^xy

2.1.3.3. (ab)^x=a^x b^x

2.1.3.4. (a/b)^x= (a^x)/(b^x)

2.1.3.5. (a^x)/(a^y)=a^x-y

2.1.4. A function of f is one-to-one if every number in its range  corresponds to exactly one number i its domain.

2.1.5. If it is not one-to-one, then their is no inverse.

2.1.6. Real Life applications

2.1.6.1. Half-life decay model

2.1.6.2. Compound interest

2.1.6.3. Doubling time growth

2.1.6.4. Population growth

2.2. Logarithmic

2.2.1. Common Log

2.2.1.1. if there is no base, the base will automatically 10

2.2.2. Natural Log

2.2.2.1. it has a base of e written as lnx

3. Basic Business Mathemetics

3.1. Simple interest

3.1.1. earned when only the original principal earns interest for a duration of the term.

3.1.2. I=Prt

3.2. Maturity value

3.2.1. sum of the principal and interest rate.

3.2.2. F=(1+rt)

3.3. Compound interest

3.3.1. type of interest that results from the periodic addition of simple interest to principal.

3.3.2. F=P(1+i)^n

3.4. Annuity -  sequence of equal payment

3.4.1. Simple - interest period is the same as the payment period.

3.4.2. General - interest period and payment period are not the same.

3.4.3. - the payments are made are made at the end of each interest period.

3.4.4. Deffered - the first payment is not made at the end of the first interest period but later.

3.5. Stocks

3.5.1. shares of ownership in a company

3.6. Bonds

3.6.1. is a loan.

3.6.2. for corporations to raised the necessary funds.

3.7. Amortization

3.7.1. dept-repayment scheme wherein the original amount borrowed is repaid by making equal payments periodically.

3.8. Outstanding principal

3.8.1. dept that are still unpaid.

4. Logic

4.1. Proposition

4.1.1. Complete declarative sentence that is either true or false.

4.2. Connective proposition

4.2.1. Conjunction

4.2.2. Disjunction

4.2.3. Implication

4.2.3.1. Hypothesis and Conclusion

4.2.3.2. logically equivalent to its contrapositive

4.2.4. Biconditional

4.3. Compound proposition

4.3.1. Tautology

4.3.1.1. always true

4.3.2. Contradiction

4.3.2.1. Always false

4.3.3. Contingency

4.3.3.1. neither tautology nor contradiction.

4.4. Valid arguement

4.4.1. Rules of inference

4.5. Fallasies

4.5.1. incorrect reasoning resembling rules of inference.

4.6. Method of proofs

4.6.1. Direct

4.6.2. Indirect

4.6.3. Existence

4.6.4. Nonexistence