and cannot be combined because their arguments differ. The arguments must be the same to rewrite trig functions. 0Find the domain and range of
sin and cos are periodic functions, meaning the outputs will repeat once the input is shifted over a full period. The unit circle is a good way to see this. No matter what real numbers we put into the function, the same pattern repeats. The output never exceeds 1 or goes below -1
Domain (), Range [-1, 1] 0Find the domain and range of
This is similar to the previous problem, but it's a bit trickier because it introduces zero into the denominator. The function is said to be undefined where the input results in a zero in the denominator. For this function, we won't have any problems when the denominator is not zero. So the domain involves any x when is not zero. is zero at any multiple of , including zero. Now, we're dividing by really tiny numbers (approaching zero). An integer divided by a very small number, is a very large number. Domain is all real numbers, but cannot equal any multiple of . where n is any integer. Range 0Graph
This is very similar to the regular cos graph, except that its period is changed, or quished." Its usual period of is now .
This is similar to the regular sin graph, but it is shifted UP by one unit. 0Graph
is very different from . Radians are used unless otherwise specified. By convention, the variable typically implies use of degrees).
has a period of , meaning that going units in x in either direction will return the same result
so this is the same graph as
The number out front here determines the amplitude. In this case, it's 3. So from minimum to maximum, this graph will cover 6 units total. Multiplying the x here by 2 changes the period from to .
The positive 4 there moves the graph over 4 units to the left. The -1 in that location means the graph is shifted down 1. This is the same graph as the previous problem, but shifted down 1 and 4 to the left.
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
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