For each of the following problems, find (a) the period, (b) the domain, and (c) the range.
For the domain, enter it x≠a±bπn, where a is the smallest positive angle, and n is an integer or fraction.
1. y=9tan(3x)
(a) Period
pi/3
(b) Domain: x≠ pi/6± 1/3πn
(c) For each of the following problems, find (a) the period, (b) the domain, and (c) the range.
For the domain, enter it x≠a±bπn, where a is the smallest positive angle, and n is an integer or fraction.
1. y=9tan(3x)
(a) Period
pi/3
(b) Domain: x≠ pi/6± 1/3πn
(c) Range
[-INF, INF]
thank you
ok sorry i posted it twice again
the actual content looks good
To find the period, domain, and range of the function y = 9tan(3x), you can follow these steps:
Step 1: Find the Period
The period of a function is the length of one complete cycle. For the tangent function, the period is π/3. This can be determined by looking at the coefficient of x inside the tangent function, which is 3. The period of tangent is π divided by the coefficient inside the function, so in this case, the period is π/3.
(a) The period of y = 9tan(3x) is π/3.
Step 2: Find the Domain
The domain of a function represents all the possible values of x that the function can take. For trigonometric functions, the domain is typically restricted because certain values of x can cause the function to be undefined. In the case of tangent, any value of x that makes the denominator of the tangent function equal to zero needs to be excluded.
The denominator of the tangent function is 3x in this case. To find the excluded values of x, we need to solve the equation 3x = π/2 + kπ, where k is an integer, to get the values of x that make the tangent function undefined. Slightly rearranging the equation, we have x = (π/2 + kπ)/3.
Since we want the smallest positive angle, the smallest positive value of k that satisfies the equation is k = 0. Plugging this value into the equation, we get x = π/6.
Therefore, the domain of y = 9tan(3x) is x ≠ π/6 ± (1/3)πn, where n is an integer or fraction.
(b) The domain of y = 9tan(3x) is x ≠ π/6 ± (1/3)πn.
Step 3: Find the Range
The range represents all the possible values of y that the function can produce. For tangent, the range is all real numbers from negative infinity to positive infinity, which means there are no limitations on the values of y.
(c) The range of y = 9tan(3x) is (-∞, ∞).
So, the final answers are:
(a) The period is π/3.
(b) The domain is x ≠ π/6 ± (1/3)πn.
(c) The range is (-∞, ∞).