When determining the domain of a function, the two main classes of functions you

Question

When determining the domain of a function, the two main classes of functions you have to worry about are rational functions (think fractions) and those functions containing a radical with an even index.   The domain of a rational function cannot contain a number that makes the denominator zero.   Why?

Give an example of a rational function that has no restrictions on the domain.   In other words the domain is all real numbers. What is the concern with radicals that have an even index?   Explain why you do not have to worry about functions containing a radical with an odd index

When determining the domain of a function, the two main classes of functions you have to worry about are rational functions (think fractions) and those functions containing a radical with an even index.   The domain of a rational function cannot contain a number that makes the denominator zero.   Why?

Give an example of a rational function that has no restrictions on the domain.   In other words the domain is all real numbers. What is the concern with radicals that have an even index?   Explain why you do not have to worry about functions containing a radical with an odd index

Explanation / Answer

domain is all values of x for which the function exists

while finding domain of rational function

we find those values that makes denominator equal to 0

as those values are excluded from its domain

because if denominator becomes 0 then function would be undefined

for example

f(x) = (x-1) / x

in this case , domain is all values of x , except x = 0

because if x = 0

the denominator would become 0 and function is undefined

rational function with no restriction on domain is

f(x) = ( x - 5 ) / 8

since there is no x term in denominator

hence, no restriction

radicals having even index

since radicals with even index should have value greater than equal to 0

if its less than 0 , then function would be complex number

so we set the inside expressiom of radical greater than equal to 0

and solve for x

this will give the domain of radical with even index

whereas domain of index with odd numbers is always real numbers , there is no restriction

unlike even radicals which cannot take negative numbers , an odd radical can take all real numbers .

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