Exploring the Function f(x) = -log₂(-x) | Properties, Domain, Range, Symmetry, and Graph

f(x) = -log₂(-x)

The given function is f(x) = -log₂(-x)

The given function is f(x) = -log₂(-x).

Let’s break down this function step by step.

1. The input of the function is denoted by x. It means you can plug in any real number or variable for x.
2. Inside the function, we have -x, which is the negative of x. This means that whatever value of x you choose, it will be negated.
3. Next, we have -log₂(-x). The symbol “log₂” represents the logarithm with base 2. Taking the logarithm allows us to find the exponent to which the base (2 in this case) should be raised to obtain the input number (-x in this case).
4. However, there is a negative sign in front of the logarithm. This means that the output of the logarithm will be negated.

To summarize, the function f(x) = -log₂(-x) takes the input, negates it, finds the base 2 logarithm of the negated value, and then negates the result again.

Now, let’s examine some properties and characteristics of this function:

1. Domain: The function f(x) is defined for all real values of x, except when x = 0. This is because the logarithm function is undefined for non-positive real numbers, and when x = 0, the input to the logarithm becomes -log₂(0), which is undefined.

2. Range: The range of the function f(x) depends on the domain of x. Since x can be any real number except 0, the range will also consist of all real numbers. The negative sign in front of the logarithm ensures that the range includes negative numbers as well.

3. Symmetry: The function f(x) is an odd function. This means that if we substitute -x in place of x, the function remains the same except for the sign in front. Specifically, f(-x) = log₂(-(-x)) = log₂(x) = -f(x). This symmetry indicates that the graph of the function is symmetric with respect to the origin.

4. Behavior as x approaches infinity: As x approaches positive infinity, -x approaches negative infinity. So, the function becomes -log₂(-∞), which is not defined in the real number system. Therefore, the function does not have a limit as x approaches infinity.

5. Behavior as x approaches negative infinity: As x approaches negative infinity, -x approaches positive infinity. So, the function becomes -log₂(∞), which is also not defined in the real number system. Hence, the function does not have a limit as x approaches negative infinity.

6. Graph: The graph of f(x) = -log₂(-x) would have a vertical asymptote at x = 0 since the logarithm is not defined for negative or zero values. It will also have symmetry with respect to the origin due to the odd nature of the function.

I hope this explanation helps you understand the given function better. Let me know if you have any further questions!

More Answers: