Calculus I

Front 1

What is important when we define a function?

Back 1

It is crucial to specify its the Domain, it’s codomain and the rule (expression) that assigns to each X that belongs to the domain of the function it’s corresponding and unique f(x).

- Domain

- codomain

- The expression

Front 2

When we are only given the expression of the function what do we assume?

Back 2

We assume that the codomain is R and the domain is the largest subset of R where the function is well-defined (the maximal Domain).

Front 3

When are two functions equal?

Back 3

To functions are only equal when:

- their expressions are the same for every x in the same domain

- they have the same domain

Front 4

What is the range?

Back 4

It is the set of all the values of f(x) that the function attains.

We can also find the range of a function by obtaining the domain of the inverse function as long as it exists.

Front 5

When does the inverse function exist?

Back 5

To invert a function, the original function needs to be injective.

Front 6

When can we generate a composite function?

Back 6

We can only generate a composite function when the range of the second belongs to the domain of the 1st.

For example:

Let f: Df —> R and g: Dg —> R, if Rf (is contained in) Dg.

Front 7

How to find the domain of a composite function,of g after f?

Back 7

The domain of a composite function is defined by:

D(g•f)={x€R: x € Df ^ f(x) € Dg}

Front 8

Definition of increasing functions

Back 8

Are real function of a real variable of is increasing in [a,b] if and only if whatever the points x1, x2 € [a,b], we have that:

F(x1) = f(x2) whenever x1

Front 9

Is a constant function considered to be an increasing function?

Back 9

Yes!

Front 10

What is is strictly increasing function? Give me a definition.

Back 10

If we have f(x1)

Front 11

What is a monotonic function?

Back 11

If a function is increasing or decreasing, the function is sad to be monotonic.

Front 12

What is a strictly monotonic function?

Back 12

If the function is strictly increasing or strictly decreasing the function is set to be strictly monotonic.

Front 13

What is an odd function?

Back 13

Is a function that respects the condition below:

f(-x)=-f(x)

Front 14

What is an even function?

Back 14

Is the function that respects the condition below:

F(x)=f(-x)

Front 15

What is the difference between an injective function and an one to one function?

Back 15

There is no difference, they are synonymous.

Front 16

What is an injective function?

Back 16

We say that f is injective if for all x,y € R, f(x) = f(y) implies that x=y.

Front 17

What is the difference between a surjective function and an onto function?

Back 17

There is no difference, they are synonymous

Front 18

What is a surjective function?

Back 18

We say that I f is surjective if its range coincides with its domain, that is, if every element of the codomain is the value (image) of some element X.

Front 19

What is a bijective function?

Back 19

A function is a bijection if it is both injective and surjective.