How can I find the domain and range of a function from the equation of that function(without using a graph of any kind)? For example, how would I find the domain and range of f(x)=3sin(2x)?
Thank you for your help!
there is a y for every x from -oo to + oo so domain is all real x
y never gets higher than 3 or lower than -3 so range is from -3 to +3
To find the domain and range of a function without using a graph, you can follow these steps:
1. Start with the equation of the function. In this example, the equation is f(x) = 3sin(2x).
2. Determine any restrictions on the variable, x, that would make the function undefined. For this specific function, there are no restrictions on x since sin(2x) is defined for all values of x.
3. For the domain, consider all possible values that x can take. The domain is the set of all valid input values for the function. In this case, the domain is all real numbers because there are no restrictions on x.
4. To find the range, think about the possible outputs of the function. The range is the set of all possible output values. For the function f(x) = 3sin(2x), the range of the sine function is typically between -1 and 1. By multiplying the sine function by 3, the range is then scaled by a factor of 3, making it between -3 and 3.
Therefore, for the given function f(x) = 3sin(2x), the domain is all real numbers, and the range is between -3 and 3.
To find the domain and range of a function given its equation, you can follow these steps:
1. Start with the definition of the function. In this case, you have f(x) = 3sin(2x).
2. Determine the restrictions on the variable, x, that would make the function undefined or meaningless. In this example, there are no explicit restrictions on x.
3. Identify any values of x that would result in the function being undefined due to division by zero or taking the square root of a negative number. However, in this example, there are no such cases.
4. If there are no explicit or implicit restrictions, then the domain is the set of all real numbers. For this function f(x) = 3sin(2x), the domain is (-∞, ∞).
5. Next, determine the range, which represents the set of all outputs or possible values of the function f(x).
a) Firstly, recall the range of the sine function, which is [-1, 1]. The amplitude of the function (3) does not affect the range.
b) Next, consider that the function f(x) = 3sin(2x) repeats itself over intervals of 2π, as the period of the sine function is 2π/2 = π.
c) Since multiplying x by 2 in f(x) = 3sin(2x) compresses the function horizontally, the range of the function will also be compressed, spanning from -3 to 3.
d) Therefore, the range of f(x) = 3sin(2x) is [-3, 3].
To summarize, for the function f(x) = 3sin(2x), the domain is (-∞, ∞) and the range is [-3, 3].