What is "Domain" and "Range" in mathematics? How can we find it from graph?
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Domain
The domain of a function is the complete set of possible values of the independent variable.
In plain English, this definition means:
The domain is the set of all possible x-values which will make the function "work", and will output real y-values.
When finding the domain, remember:
The denominator (bottom) of a fractioncannot be zeroThe number under a square root signmust be positive in this section
The domain of this function is x≥−4\displaystyle{x}\ge-{4}x≥−4, since x cannot be less than −4\displaystyle-{4}−4. To see why, try out some numbers less than −4\displaystyle-{4}−4 (like −5\displaystyle-{5}−5 or −10\displaystyle-{10}−10) and some more than −4\displaystyle-{4}−4 (like −2\displaystyle-{2}−2 or 8\displaystyle{8}8) in your calculator. The only ones that "work" and give us an answer are the ones greater than or equal to −4\displaystyle-{4}−4. This will make the number under the square root positive.
Notes:
The enclosed (colored-in) circle on the point (−4,0)\displaystyle{\left(-{4},{0}\right)}(−4,0). This indicates that the domain "starts" at this point.
How to find the domain
In general, we determine the domain of each function by looking for those values of the independent variable (usually x) which we are allowed to use. (Usually we have to avoid 0 on the bottom of a fraction, or negative values under the square root sign).
Range
The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain.
In plain English, the definition means:
The range is the resulting y-values we get after substituting all the possible x-values.
How to find the range
The range of a function is the spread of possible y-values (minimum y-value to maximum y-value)Substitute different x-,values into the expression for y to see what is happening. (Ask yourself: Is y always positive? Always negative? Or maybe not equal to certain values?)Make sure you look for minimum and maximum values of y .Draw a sketch! In math, it's very true that a picture is worth a thousand words.
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The domain of a function is the complete set of possible values of the independent variable.
In plain English, this definition means:
The domain is the set of all possible x-values which will make the function "work", and will output real y-values.
When finding the domain, remember:
The denominator (bottom) of a fractioncannot be zeroThe number under a square root signmust be positive in this section
The domain of this function is x≥−4\displaystyle{x}\ge-{4}x≥−4, since x cannot be less than −4\displaystyle-{4}−4. To see why, try out some numbers less than −4\displaystyle-{4}−4 (like −5\displaystyle-{5}−5 or −10\displaystyle-{10}−10) and some more than −4\displaystyle-{4}−4 (like −2\displaystyle-{2}−2 or 8\displaystyle{8}8) in your calculator. The only ones that "work" and give us an answer are the ones greater than or equal to −4\displaystyle-{4}−4. This will make the number under the square root positive.
Notes:
The enclosed (colored-in) circle on the point (−4,0)\displaystyle{\left(-{4},{0}\right)}(−4,0). This indicates that the domain "starts" at this point.
How to find the domain
In general, we determine the domain of each function by looking for those values of the independent variable (usually x) which we are allowed to use. (Usually we have to avoid 0 on the bottom of a fraction, or negative values under the square root sign).
Range
The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain.
In plain English, the definition means:
The range is the resulting y-values we get after substituting all the possible x-values.
How to find the range
The range of a function is the spread of possible y-values (minimum y-value to maximum y-value)Substitute different x-,values into the expression for y to see what is happening. (Ask yourself: Is y always positive? Always negative? Or maybe not equal to certain values?)Make sure you look for minimum and maximum values of y .Draw a sketch! In math, it's very true that a picture is worth a thousand words.
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