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  • Use functional notation to evaluate a function.
  • Determine the domain and range of a function.
  • Draw the graph of a function.
  • Find the zeros of a function.
  • Recognize a function from a table of values.
  • Make new functions from two or more given functions.
  • Describe the symmetry properties of a function.

In this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Most of this material will be a review for you, but it serves as a handy reference to remind you of some of the algebraic techniques useful for working with functions.


Given two sets A and B , a set with elements that are ordered pairs ( x , y ) , where x is an element of A and y is an element of B , is a relation from A to B . A relation from A to B defines a relationship between those two sets. A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the input ; the element of the second set is called the output . Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.


A function     f consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The set of inputs is called the domain    of the function. The set of outputs is called the range    of the function.

For example, consider the function f , where the domain is the set of all real numbers and the rule is to square the input. Then, the input x = 3 is assigned to the output 3 2 = 9 . Since every nonnegative real number has a real-value square root, every nonnegative number is an element of the range of this function. Since there is no real number with a square that is negative, the negative real numbers are not elements of the range. We conclude that the range is the set of nonnegative real numbers.

For a general function f with domain D , we often use x to denote the input and y to denote the output associated with x . When doing so, we refer to x as the independent variable    and y as the dependent variable    , because it depends on x . Using function notation, we write y = f ( x ) , and we read this equation as y equals f of x . For the squaring function described earlier, we write f ( x ) = x 2 .

The concept of a function can be visualized using [link] , [link] , and [link] .

An image with three items. The first item is text that reads “Input, x”. An arrow points from the first item to the second item, which is a box with the label “function”. An arrow points from the second item to the third item, which is text that reads “Output, f(x)”.
A function can be visualized as an input/output device.

Questions & Answers

The f'(4)for f(x) =4^x
Alice Reply
I need help under implicit differentiation
Uchenna Reply
how to understand this
What is derivative of antilog x dx ?
Tanmay Reply
what's the meaning of removable discontinuity
Brian Reply
what's continuous
an area under a curve is continuous because you are looking at an area that covers a range of numbers, it is over an interval, such as 0 to 4
using product rule x^3,x^5
please help me to calculus
World Reply
may god be with you
Luke 17:21 nor will they say, See here or See there For indeed, the kingdom of God is within you. You've never 'touched' anything. The e-energy field created by your body has pushed other electricfields. even our religions tell us we're the gods. We live in energies connecting us all. Doa/higgsfield
if you have any calculus questions many of us would be happy to help and you can always learn or even invent your own theories and proofs. math is the laws of logic and reality. its rules are permanent and absolute. you can absolutely learn calculus and through it better understand our existence.
ya doubtless
help the integral of x^2/lnxdx
also find the value of "X" from the equation that follow (x-1/x)^4 +4(x^2-1/x^2) -6=0 please guy help
Use integration by parts. Let u=lnx and dv=x2dx Then du=1xdx and v=13x3. ∫x2lnxdx=13x3lnx−∫(13x3⋅1x)dx ∫x2lnxdx=13x3lnx−∫13x2dx ∫x2lnxdx=13x3lnx−19x3+C
itz 1/3 and 1/9
now you can find the value of X from the above equation easily
Pls i need more explanation on this calculus
usman from where do you need help?
thanks Bilal
integrate e^cosx
-sinx e^x
Do we ask only math question? or ANY of the question?
Levis Reply
How do i differentiate between substitution method, partial fraction and algebraic function in integration?
you just have to recognize the problem. there can be multiple ways to solve 1 problem. that's the hardest part about integration
we asking the question cause only the question will tell us the right answer
find integral of sin8xcos12xdx
Levis Reply
don't share these childish questions
well find the integral of x^x
bilal kumhar you are so biased if you are an expert what are you doing here lol😎😎😂😂 we are here to learn and beside there are many questions on this chat which you didn't attempt we are helping each other stop being naive and arrogance so give me the integral of x^x
Levis I am sorry
Bilal it okay buddy honestly i am pleasured to meet you
x^x ... no anti derivative for this function... but we can find definte integral numerically.
thank you Bilal Kumhar then how we may find definite integral let say x^x,3,5?
evaluate 5-×square divided by x+2 find x as limit approaches infinity
Michagaye Reply
i have not understood
The answer is 0
I just dont get it at all...not understanding
0 baby
The denominator is the aggressive one
wouldn't be any prime number for x instead ?
or should I say any prime number greater then 11 ?
just wondering
I think as limit Approach infinity then X=0
ha hakdog hahhahahaha
No Reply
ha hamburger
Fond the value of the six trigonometric function of an angle theta, which terminal side passes through the points(2x½-y)²,4
albert Reply
What's f(x) ^x^x
Emeka Reply
What's F(x) =x^x^x
are you asking for the derivative
that's means more power for all points
if your asking for derivative dy/dz=x^2/2(lnx-1/2)
iam sorry f(x)=x^x it means the output(range ) depends to input(domain) value of x by the power of x that is to say if x=2 then x^x would be 2^2=4 f(x) is the product of X to the power of X its derivatives is found by using product rule y=x^x introduce ln each side we have lny=lnx^x =lny=xlnx
the derivatives of f(x)=x^x IS (1+lnx)*x^x
So in that case what will be the answer?
nice explanation Levis, appreciated..
what is a maximax
Chinye Reply
A maxima in a curve refers to the maximum point said curve. The maxima is a point where the gradient of the curve is equal to 0 (dy/dx = 0) and its second derivative value is a negative (d²y/dx² = -ve).
what is the limit of x^2+x when x approaches 0
Dike Reply
it is 0 because 0 squared Is 0
simply put the value of 0 in places of x.....
the limit is 2x + 1
the limit is 0
limit s x
The limit is 3
Leo we don't just do like that buddy!!! use first principle y+∆y=x+∆x ∆y=x+∆x-y ∆y=(x+∆x)^2+(x+∆x)-x^2+x on solving it become ∆y=3∆x+∆x^2 as ∆x_>0 limit=3 if you do by calculator say plugging any value of x=0.000005 which approach 0 you get 3
find derivatives 3√x²+√3x²
Care Reply
3 + 3=6
How to do basic integrals
dondi Reply
the formula is simple x^n+1/n+1 where n IS NOT EQUAL TO 1 And n stands for power eg integral of x^2 x^2+1/2+1 =X^3/3

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