which function has a domain of all real numbers?

Question

A. y=cos x

B. y=sec x

C. y=1/ sin x

D. y= tan x

Answer 1

Hi Raul,

 

The domain refers to all of the possible x-values that, when input into an equation, will yield a valid output (y-value).  In order for the domain to contain "all real numbers," there can be no x-values for which the equation is undefined.  A hint I can provide is to look at each equation that is given and identify if any are undefined for certain x-values.  Remember that when using fractions, we cannot divide by zero, so any equation would be undefined where plugging in a certain x-value would result in a zero in the denominator.  Graphing these equations will also help: vertical asymptotes are x-values for which the equation is undefined.

 

For example, Answer Choice C: y = 1/sin x.  Because we cannot divide by zero, this function will be undefined whenever the denominator, sin x = 0.  Sin x = 0 whenever x is a multiple of π (pi).  Graphing the full equation, y = 1/sin x (which is also the same as y = csc x), shows vertical asymptotes wherever x is equal to a multiple of π.  Answer Choice C is undefined for certain x-values and so does not have a domain of all real numbers.