Domain and Range of a Function
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Domain and Range of a Function
In mathematics, domain is an important concept in a function. Domain is defined as a
set of input value or arguments. The set of all the output values is called as a range.
The range is an interval among the uppermost number to the minor number. In
mathematics, the domain and range function is one of the divisions.
How to Find the Domain and Range of a Function?
For any function y=f(x), the domain is the set of al the possible values of x, where x is the
independent variable. Range of a function is the set of al the corresponding values of y, for
every value of x.
Finding Domain and Range of a Function
Given below are some of the examples on finding the domain and range of a function.
Domain and Range of a Quadratic Function
Know More About : Graphing Rational Functions
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A quadratic function can be expressed as f(x) = ax2 + bx + c, where a, b and c are the
constants and a a 0. The domain of a quadratic function is the set of all values that makes
the given definition true and similarly the set of all f(x) or y values that makes the function true
is cal ed the range of the function.
Solved Example
Question: Find the domain and range of the function f(x) = x2 + 5
Solution:
The above function denotes the polynomial of degree 2, which is quadratic. The graph of the
above polynomial function is a parabola which opens upwards. For every value of x, there is a
corresponding value of y. On the real number line, the x can take any value.
Let us restrict the domain such that 3x3, x R
[set of all real numbers ]
Meaning, Domain = { x : x
[ 3, 3 ] , x
R}
The corresponding values of y are,
f(3) = (3)2 + 5 = 9 + 9 = 14
f(2) = (2)2 + 5 = 4 + 5 = 9
f(1) = (1)2 + 5 = 1+ 5 = 6
f(0)= (0)2 + 5 = 5
f(1) = 12 + 5 = 1 + 5 = 6
Learn More : Volume of a Cone Formula
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f(2) = 22 + 5 = 4 + 5 = 9
f(3) = 32 + 5 = 9 + 5 = 14
Hence the set of coordinates on the graph of f(x) = x2 + 5 are (3, 14), (2, 9), (1, 6), (0,5),
(1,6), (1,9)
Domain and Range of Logarithmic Functions+
If the logarithmic function is in the form of f(x) = loga x, then the domain of this is (0, ) and
range is (, ). And, if the function is in the form of f(x) = loga(x  k), then the domain is (k ,
) and range is (, ).
Solved Example
Question: Find the Domain and Range of the function y = log(x + 4)  6
Solution:
We have the function f(x) = log(x + 4)  6 [common logarithm, which is of base 10 ]
Step 1:
Domain : Since log of 0 or log of negative is not defined , (x + 4) > 0 => x > 4
Domain consist of all real numbers > 4
Domain = { x : x
( 4, )
Step 2 : When x > 4, say x = 3, y = log(3 + 4)  6 = log 1  6 = 0  6 = 6
When x = 2, y = log (2 + 4)  6 = log 2  6 = 1.414  6 = 4.586
As x increases for values > 4, y also increases from  to +.
Range = { y : y
( , ) }
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