f(x) = log2(x – 3) + 1. How do I find the Domain and Range?
you know that log(u) has domain (0,∞) and range (-∞,+∞)
So, log(x-3) has domain (x-3)>0 or x>3.
That is (3,∞)
range is still the same. Adding 1 to ∞ makes no difference.
To find the domain and range of the function f(x) = log2(x - 3) + 1, we need to consider the restrictions and limitations on the input and output values of the logarithmic function.
1. Domain:
The domain of a logarithmic function is determined by the values inside the logarithm. In this case, the expression inside the logarithm is x - 3. For the logarithm function to be defined, the expression x - 3 must be greater than 0, since the logarithm with a base of 2 is undefined for non-positive numbers.
So, we set x - 3 > 0 and solve for x:
x - 3 > 0
x > 3
Therefore, the domain of the function is all real numbers greater than 3, or in interval notation: (3, ∞).
2. Range:
The range of a logarithmic function is the set of all possible output values. The range of the function f(x) = log2(x - 3) + 1 can be determined by understanding the behavior of logarithmic functions.
Since the base of the logarithm is 2, the function will only take positive values. The logarithm of a positive number is always a real number, so the range of f(x) is all real numbers.
In interval notation, the range can be represented as (-∞, ∞).
To find the domain and range of the function f(x) = log2(x – 3) + 1, we need to consider the restrictions on the input values (x) and the resulting output values (f(x)).
The domain of a function refers to the set of all possible input values for which the function is defined. In this case, the function f(x) = log2(x – 3) + 1 involves the logarithm function, which is only defined for positive real numbers. Therefore, for the given function, the expression (x – 3) must be greater than zero. Solving this inequality, we have:
x – 3 > 0
Adding 3 to both sides, we get:
x > 3
Hence, the domain of the function is all real numbers greater than 3, or expressed in interval notation, (3, ∞).
Now let's move on to finding the range. The range of a function refers to the set of all possible output values resulting from the given set of input values. In this case, since the base of the logarithm is 2, the function f(x) = log2(x – 3) + 1 will take on all possible real values.
To find the range, we consider the behavior of the logarithm function. As the input values (x – 3) approach positive infinity, the logarithm function will increase without bound. Adding 1 to this result shifts the graph vertically upward by 1 unit. Therefore, the range of the given function f(x) is all real numbers greater than or equal to 1, or in interval notation, [1, ∞).
In summary:
Domain: (3, ∞)
Range: [1, ∞)