Analyzing The Function F(X) = -Log₂(-X): Domain, Intercepts, Intervals, Concavity, And Asymptotes

f(x) = -log₂(-x)

Reflect base-2 log across x-axis and y-axis

To analyze the function f(x) = -log₂(-x), we need to consider the domain, x-intercept, y-intercept, increasing and decreasing intervals, concavity, and any asymptotes.

1. Domain: The domain of f(x) is (-∞, 0). This is because the expression -log₂(-x) is only defined for negative values of x, and the base of the logarithm must be greater than 0.

2. x-intercept: To find the x-intercept, we need to set f(x) = 0 and solve for x. However, we cannot evaluate the function at x = 0 or any positive value, since the logarithm of a non-negative number is not defined. Therefore, there are no x-intercepts.

3. y-intercept: To find the y-intercept, we need to evaluate f(x) at x = 0. However, 0 is not in the domain of the function, so there is no y-intercept.

4. Increasing and decreasing intervals: To determine where the function f(x) is increasing or decreasing, we first take the derivative: f'(x) = 1/(x*log(2)). This derivative is defined for all x in the domain of f(x), which is (-∞, 0). Setting f'(x) = 0, we find that there is a critical point at x = -1. We then construct a sign chart for f'(x) using test points and find that f(x) is decreasing on (-∞, -1) and increasing on (-1, 0).

5. Concavity: To determine where the function f(x) is concave up or concave down, we take the second derivative: f”(x) = -1/(x²*log(2)). This derivative is defined for all x in the domain of f(x), but it is negative for all x in the domain. Therefore, f(x) is concave down for all x in the domain.

6. Asymptotes: To find the vertical asymptote, we need to determine where the denominator in the expression 1/(x*log(2)) is equal to 0. This occurs when x = 0, so the vertical asymptote is x = 0. To find the horizontal asymptote, we take the limit as x approaches negative infinity: lim(x→-∞) -log₂(-x) = -∞. Therefore, there is no horizontal asymptote.

In summary, the function f(x) = -log₂(-x) has a domain of (-∞, 0), no x-intercepts, no y-interceptors, a vertical asymptote at x = 0, is decreasing on (-∞, -1) and increasing on (-1, 0), is concave down for all x in its domain, and has no horizontal asymptotes.

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