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functions and function notation
10/05/2023
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functions and function notation. WHAT IS A FUNCTION ? A function is the relationship between inputs and outputs, with each input having ONE output. HOWEVER if an input gives more than one output, it is NOT a function. A The input the domain The output = the range when functions are drawn graphically, the domain is x, and the range is y Although a function can only have one output for each input, the input can be singular or a set of inputs. = This means that you can have: • A one to one function. A many to one function You can view functions similar to a pizza maker: •x is your pizza base •The input are the toppings you ask for on your pizza • the output is the pizza you ordered. for example Input (Domain) Output (range) •Cheese •Peperoni Mushrooms (with one input and one output) (with many inputs which can give one output). f(x) = 2x input value = -4 f(-4)= 2(-4) f(-4)= -8 →X (a Simpler method of describing a function without using a lengthy written explanation.) However... Functions are actually written using function notation, Like this: f(x) = 2x or fixx² Both f(x) and fix mean the same thing: 'f of x' However, these functions input values have not yet been put in, So, say I wanted to put the input value of -4 into these functions, it would look like this fixx² input value = -4 f(x) = x² f(-4)=(-4)² f(-4)= 16 BUT, if we...
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Alternative transcript:
wanted to represent the functions graphically, we would do this: f(x) = 2x fixx² Remember the input (domain) = x the output (range) =y So these would look like у = 2х These graphs represent their funtions, but also clearly show the difference one-to-one functions and the many-to-one functions that we between the discussed earlier. •As f(x) = 2x shown graphically looks like this. y=x² 4- 1 -2 It is clear that if you wanted to get an output, y value of 4, the only input value that you could use would be 2. This shows that f(x)=2x is a one to one function. as only one input can give a specified output. • As fix →x² shown graphically, looks like this: It is clear that if you wanted to get an output, y value of 4, there is a choice of two input (x) values you could use either -2 or 2. This shows that fix →x² is a many to one function as there is a choice of different inputs which create the Same one output. @f(x)=x+2 (b) exampl Are the following functions? = es- remember, to be a function it must: • Have ONE or MANY inputs. • Have ONLY ONE output. Lets draw the graph to see: f(x)=x+2 y = x + 2 y intercept = 2 gradient-1 As you can see, any input (a) only has ONE output (y) YES This is a one to one function. remember, to be a function it must: •Have ONE or MANY inputs •Have ONLY ONE output. As you BUT, more than one input (x) can give the same ONE output (y) can see, any in put (x) only has ONE output (y) YES: this is a many to one function. - practice Is f(x) = sin(x) a function? Is f(x) = √x a function? PRACTICE ANSWERS (1) Is f(x) = sin(x) a function? remember, to be a function it must: •Have ONE or MANY inputs • Have ONLY ONE output. Lets draw the graph to see... AMA = YES: this is a many to one function (2) Is f(x) = √√5c a function ? remember, to be a function it must: • Have ONE or MANY inputs • Have ONLY ONE output. = As you can see, any input (a) only has ONE output (y). BUT, more than one input (x) can give the same ONE output (y) Lets use the input (4): f(4)=√4 √4 = 2 or 2 & this means that there are 2 outputs when in an input you Or, if you draw the graph... If this means that there are 2 outputs (y) when in an input (x) you NO! This is NOT a function ! put put