Function Tutorial

8/2/2019 Function Tutorial

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FunctionsFunctions

By SoundaraBy Soundara


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Definition:

A function f is a mathematical expression which takesinput 'x' and produce corresponding output f(x). For anyfunction, each input x gives exactly one out put f(x).

Afunctionconsists of three things;i) A set called the domainii) A set called the range

iii) A rule which associates each element of the domainwith a unique element ofthe range.


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Function notation: y = f(x) where x is theindependent variable and y is the dependent variable.

We normally write functions as: f(x) and read this as"function f of x".

We can use other letters for functions, like g(x) or y(x).


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DOMAIN:

The set of all inputs that a function accepts is calleddomain.

Example: if a function f(x) can be given the values x ={1,2,3,...} then {1,2,3,...} is the domain.


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RANGE:

The set of all output values of a function is called as range.

Example: if the function f(x) = x^2 is given the values x ={1,2,3,...} then its range will be x^2 = {1,4,9,...}


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1. Find the domain and range for the function f(x) = x^2 + 2.Also plot the graph of this function.


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Solution:

The function f(x) = x^2 + 2 is defined for all real values of x

(because there are no restrictions on the value of x).

Note: The set of real numbers includes all integers, positiveand negative; all fractions; and the irrational numbers.

Hence, the domain of f(x) is "all real values of x".


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Since x^2 is never negative, x^2 + 2 is never less than 2

Hence, the range of f(x) is "all real numbers f(x) 2".

To Plot the graph let us find some of the f(x) values by havingthe values as x = .... ,-2,-1,0,1,2, ...

x -2 -1 0 1 2

f(x) = x^+2 (-2)^2+2 = 6 (-1)^2+2 = 3 0+2 = 2 (1)^2+2 = 3 (2)^2 +2 = 6


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We can see that x can take any value in the graph, but theresulting f(x) values are greater than or equal to 2.


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2. Determine the domain and range of the given

function:

f(x) = sqrt ( x+4)

Solution:

Note: The values under a square root sign must bepositive.The domain of the function is x 4, since x cannottake values less than 4. which is [-4, infinity)And the range will be [0, infinity).


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3. Plot the graph for the function f(x) = sqrt(x) -2.


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Solution:

Here x can take only the non negative real number. Negativevalues are not possible for sqrt. Hence the domain of thefunction will be [0, infinity).

Therefore, x = 0,1,2,...

x f(x) = sqrt(x) - 2

0 0 - 2 = -2

1 1 - 2 = -12 Sqrt(2) 2 = -0.6

3 Sqrt(3) - 2 = -0.3

4 Sqrt(4)-2 = 0


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This is the graph of the function f(x) = sqrt(x) -2


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Composition Of Functions:

The term "composition of functions" refers to thecombining of functions in a manner where the outputfrom one function becomes the input for the next

function.

The notation used for composition is:

(fog)(x) = f(g(x)).

and is read "f composed with g of x" or "f of g of x".


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Example:

1. Find the composition function (fo g) (x)

f(x) = x + 1 , g(x) = 3x

Solution:(fog)(x) = f(g(x))= f(3x) {since g(x) = 3x}= (3x) +1{f(x) = x+1, here in the place of x we have 3xsince we need to replace x by 3x}.(fog)(x) = 3x + 1


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2. Find the composition function (fo g) (x) and its

domain.

f(x) = x2 + 1 , g(x) = sqrt(2 x)

Solution:

(fog)(x) = f(g(x))= f(sqrt(2x))= (sqrt(2x)^2 +1)= 2x +1


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Exercise:

1. Determine the range of the functionf(x) = x^2 - 3.

2. Plot the graph of the functionf(x) = |x+6|.

3.Find the composition function (fo g) (x)

f(x) = sqrt(-x + 1) , g(x) = x2 - 8

4.Find, if possible, (fo g) (-2).

f(x) = 2x + 1 , g(x) = x2


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