DU5 FUNCTIONS

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It is a correspondence between two sets, A and B.

For each element in A, it assigns one, and only one, element in B.

1. APPLICATION

f : A → B

x → y = f (x)

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Being D any non-null subset of R, a real function is any application from D to R.

Tables and graphs are used to represent them.

2. REAL FUNCTION

f: D ⊆ R → R

x → y = f (x)

x y-2 4-1 10 01 12 4

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It is a set of independent variables for which a function is defined.

There are two kinds of domains:

◦ Continuous: the function f is defined for an

interval.

◦ Discrete: the function f is only defined for some

isolated values.

3. DOMAIN

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Their domain is all R.

3.1 Polynomial functions

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3.2 Rational functions

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Their domain depends on n:

◦ If n is odd,

◦ If n is even,

3.3 Irrational functions (with square roots)

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3.4 Exponential functions

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f ( x ) = loga [g ( x )], where a > 0, a ≠1.

Dom(f )={x ∈Dom(g ) / g( x )> 0}

3.5 Logarithmic functions

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The range of a function is the set of values that the variable y can take.

Examples:

◦ Constant f.:

◦ Identity f.:

◦ Squaring f.:

◦ Cubing f.:

4. RANGE

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If for all , then the function is EVEN, symmetric to the Y axis.

If for all , then the function is ODD, symmetric to the origin.

It can also be symmetric to the X axis.

5. SYMMETRY

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Continuous function: a function without gaps.◦ Polynomial functions◦ Exponential functions◦ …

Non continuous function: a function with gaps.◦ Rational functions◦ …

6. CONTINUITY

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Essentially increasing:

Increasing:

7. MAXIMUMS & MINIMUMS7.1 Increasing and decreasing functions

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Essentially decreasing:

Decreasing:

7.1 Increasing and decreasing functions

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7.2 Relative points. Maximums & minimums.

Relative maximum: (0, 0)

Absolute minimums: (-1,-4) and (1,-4)

 

 

Absolute minimum: (0,-4)

Relative minimum: (1, 2)

Relative maximum: (-1,-2)It has no maximums or minimums.

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8. ASYMPTOTES

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The function F is said to be a periodic function with period T, where T≠0, if:

◦ T is the smallest real number that fulfils those two

conditions.

9. PERIODICITY

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9. PERIODICITY

Example: π

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LINEAR FUNCTIONS:• Their equation is .• m is the slope of the line and it cuts the Y axis

at the point (0,n)• If , the function is increasing, and if it is

decreasing.

10. FUNDAMENTAL FUNCTIONS

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QUADRATIC FUNCTIONS: Their equation is , where . The parabola’s vertex is at the point where

If the function is similar to a U. If not, it is similar to a .

10. FUNDAMENTAL FUNCTIONS

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K/X TYPE FUNCTIONS: is not in the domain. It doesn’t cut the X axis, because for every x. It is an odd symmetry function.

10. FUNDAMENTAL FUNCTIONS

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EXPONENTIAL FUNCTIONS: Their equation is , where a>0 and . They do not cut the X axis, because the equation has

no real solutions. They cut the Y axes at the point .

LOGARITHMIC FUNCTIONS: Their equation is , where a>0 and a≠1. They cut the X axis at the point (1, 0). They go through the point (a, 1).

10. FUNDAMENTAL FUNCTIONS

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TROGONOMETRIC FUNCTIONS:

10. FUNDAMENTAL FUNCTIONS

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PICE-WISE DEFINED FUNCTIONS:

10. FUNDAMENTAL FUNCTIONS

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(1/1/

(f/g)1/

11. OPERATIONS WITH FUNCTIONS11.1 +, -, * and /.

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Being and two functions of real variables with domains and domains respectively, and with ; the compose function is:

11.2 Composition of functions

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