Let S={-1,0,2,4,7} . Find f(S) if a) f(x)=1 . b) f(x)=2 x+1 c) f(x)=⌈ x / 5⌉

Let S={-1,0,2,4,7} . Find f(S) if a) f(x)=1 . b) ...

Key Concepts

Floor Function

The floor function, represented as ?x?, maps a real number to the largest integer less than or equal to that number. It plays a crucial role in quantitative disciplines when discretizing continuous variables, and is instrumental in various branches of mathematics including number theory and analysis.

Linear Function

A linear function, typically expressed in the form f(x) = mx + b, represents a straight-line relationship between the input and the output. Such functions are fundamental in algebra and serve as a stepping stone to more complex topics, providing insight into proportional relationships and rate of change.

Ceiling Function

The ceiling function, denoted as ?x?, rounds any real number up to the smallest integer that is not less than the number. It is widely used when discrete values are required from continuous inputs, and understanding it helps in solving problems involving rounding and partitioning.

Function Evaluation on a Set

This concept involves applying a function to every element of a given set. It requires taking each individual element in the domain and computing its image under the function, thereby forming a new set of outputs. This process is central in understanding mappings and transformations in mathematics.

Constant Function

A constant function is one that returns the same value no matter what input is provided. It is useful in illustrating a basic type of function where all elements in the domain are mapped to a single fixed element in the codomain, displaying properties like lack of variation and simple behavior.

Transcript

00:01 Okay, so all we need to do in this question is to find the set f as a function of s so all we really need to do is we just need to substitute the point in so for f as a function of one f of x is equal to one f of minus one is just one f of zero is equal to one f of two is equal to one f of four is equal to one f of seven is equal to one so really simply a f of the function of s is just a set of 1.

00:35 Very simple.

00:37 Now for 2, for b, for b, f as a function of x is equal to 2x plus 1.

00:49 F of minus 1 is minus 2 plus 1 gives you minus 1.

00:56 F of 0 gives you 2 times 0 plus 1 gives you 1.

01:00 F of 0 gives you 1.

01:03 Plus 1 gives you 5, f of 4 gives you 2 times 4 is 8 plus 1 gives you 9 and finally f of 7 is 2 times 7 gives you 14 plus 1 is 15.

01:17 So then your f as a function of s is then minus 1, 1, 15.

01:27 So one thing that you may want to note when you're writing these sets, it doesn't really something to keep in mind.

01:36 Doesn't really matter.

01:37 And see, f as a function of x is equal to ceiling of x divided by five.

01:45 So f of minus one is equal to ceiling of negative one over five, which is just ceiling of negative point two.

01:57 So that's just zero.

01:59 F of zero is ceiling of 0 divided by 5 which is 0 f of 2 ceiling of 2 divided by 5 which is ceiling of 0 .4 so that's just 0 f of 4 is equal to ceiling oh sorry this should be 1 sorry it's 1 ceiling of 4 divided by 5 is ceiling of 0 .8 so that's just 1 f of 0 .8 so that's just 1 f of 7, 7, which is ceiling of 7 divided by 5.

02:38 So that's 1 .4.

02:42 So that's just 2.

02:43 So that's just 2, yeah.

02:45 So then f as a function of s is just simply 0, 1 and 2...

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