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19 Cards in this Set
- Front
- Back
What is important when we define a function? |
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 |
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When we are only given the expression of the function what do we assume? |
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). |
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When are two functions equal? |
To functions are only equal when: - their expressions are the same for every x in the same domain - they have the same domain |
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What is the range? |
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. |
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When does the inverse function exist? |
To invert a function, the original function needs to be injective. |
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When can we generate a composite function? |
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. |
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How to find the domain of a composite function,of g after f? |
The domain of a composite function is defined by: D(g•f)={x€R: x € Df ^ f(x) € Dg} |
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Definition of increasing functions |
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 |
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Is a constant function considered to be an increasing function? |
Yes! |
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What is is strictly increasing function? Give me a definition. |
If we have f(x1) |
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What is a monotonic function? |
If a function is increasing or decreasing, the function is sad to be monotonic. |
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What is a strictly monotonic function? |
If the function is strictly increasing or strictly decreasing the function is set to be strictly monotonic. |
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What is an odd function? |
Is a function that respects the condition below: f(-x)=-f(x) |
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What is an even function? |
Is the function that respects the condition below: F(x)=f(-x) |
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What is the difference between an injective function and an one to one function? |
There is no difference, they are synonymous. |
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What is an injective function? |
We say that f is injective if for all x,y € R, f(x) = f(y) implies that x=y. |
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What is the difference between a surjective function and an onto function? |
There is no difference, they are synonymous |
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What is a surjective function? |
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. |
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What is a bijective function? |
A function is a bijection if it is both injective and surjective. |