what would be the domain anbd range for 3-log4(5-x)
The domain of the function 3-log4(5-x) consists of all real numbers such that the expression inside the logarithm, (5-x), is positive.
Setting 5-x > 0 and solving for x, we get x < 5. Therefore, the domain is x < 5.
The range of the function is all real numbers since the logarithm function can take on any real value.
To find the domain and range of the function f(x) = 3-log₄(5-x), we can consider two aspects: the domain and the range.
1. Domain:
The domain of a function represents all possible values that x can take in order for the function to be defined. In this case, we need to ensure that the expression inside the logarithm function is greater than zero, as logarithms are only defined for positive values.
5 - x > 0
Solving the inequality for x, we get:
x < 5
So the domain of the given function is all real numbers less than 5.
Domain: (-∞, 5)
2. Range:
The range of a function represents all possible values that the function can output. For logarithmic functions, the range is typically all real numbers.
Range: (-∞, +∞)
Therefore, for the function f(x) = 3-log₄(5-x),
Domain = (-∞, 5)
Range = (-∞, +∞)
To determine the domain and range of the function f(x) = 3-log4(5-x), we need to consider the limitations and possible values for x.
Domain:
The domain of a function typically refers to the set of all possible input values (x-values) for which the function is defined. In this case, we need to look at the expression inside the logarithm.
In the given function, we have log base 4, which means the argument (5-x) must be greater than zero since the logarithm is undefined for non-positive values. Therefore, the condition 5-x > 0 must be satisfied:
5 - x > 0
x < 5
Hence, the domain of the function is all real numbers x less than 5, expressed in interval notation as (-∞, 5).
Range:
The range of a function refers to the set of all possible output values (y-values) that the function can produce. In this case, the range will depend on the behavior of the logarithm and the constant multiplier.
Since log4(5-x) is always negative (as long as x < 5), the function f(x) = 3-log4(5-x) will always be positive. Therefore, the range is all positive real numbers, expressed in interval notation as (0, +∞).