Find the domain and range of the function:
g(x)=-2[2x]+1
What do those brackets mean? I'm pretty sure it's not a typo or anything because other functions on my work have regular parentheses instead of brackets.
Thank you!
IT means absolute value. So the domain of x is all x, the range of the function is from 1 to negative infinity.
The brackets in the function g(x) = -2[2x] + 1 indicate the use of the floor function. The floor function, denoted as [x], returns the largest integer less than or equal to x. In this case, it means that the value inside the brackets is rounded down to the nearest integer.
To find the domain and range of the function g(x), we will consider the restrictions on the input values (domain) and the output values (range).
Domain:
Since the function involves the floor function [2x], the values of x that make the floor function undefined need to be considered. The floor function is undefined when the input is not a real number or when 2x is not an integer.
In this case, we have a linear function inside the brackets, 2x. This means that any real number can be input into the function without restrictions. Therefore, the domain of g(x) is all real numbers (-∞, ∞).
Range:
To find the range, we need to consider the possible output values. The floor function rounds the given value down to the nearest integer.
For the function g(x) = -2[2x] + 1, the floor function only affects the value of 2x. The largest integer less than or equal to 2x will always be found between two consecutive integers. This means that the output of the floor function will always be one of the integer values between those two consecutive integers.
So, the output of the floor function can be any integer. However, when multiplying by -2 and adding 1, we will obtain a range of values that consists of all even integers plus 1.
Therefore, the range of g(x) is all odd integers plus 1, or { ..., -3, -1, 1, 3, ... }.
In summary, the domain of g(x) is all real numbers (-∞, ∞), and the range is all odd integers plus 1, or { ..., -3, -1, 1, 3, ... }.