Exploring Properties and Evaluation of the Logarithmic Function f(x) = log₂(x+2) – 2

f(x) = log₂(x+2)-2

To understand and evaluate the function f(x) = log₂(x+2) – 2, let’s break it down step by step:

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To understand and evaluate the function f(x) = log₂(x+2) – 2, let’s break it down step by step:

1. The function is defined as f(x) = log₂(x+2) – 2.
– The logarithm function is denoted as log. In this case, we have a logarithm base 2, indicated by log₂.
– The argument of the logarithm function is (x+2), meaning you need to add 2 to the input value of x before taking the logarithm.

2. Evaluating the function involves plugging in a specific value for x and performing the necessary calculations.
– For example, let’s calculate f(4):
f(4) = log₂(4+2) – 2
= log₂(6) – 2
≈ 2.58496 – 2
≈ 0.58496

3. The function f(x) = log₂(x+2) – 2 has several important properties:
– Domain: The function is defined for any value of x where the argument of the logarithm, (x+2), is positive. So the domain is x > -2.
– Range: The range of the function is all real numbers because the logarithm function can output any real number.
– Asymptote: There is a vertical asymptote at x = -2 since log₂(x) is undefined for negative values, and adding 2 to x does not change that fact.
– Intercept: To find the x-intercept, set f(x) = 0 and solve for x. Here, log₂(x+2) – 2 = 0. By rearranging, we have log₂(x+2) = 2. Converting to exponential form, we get x+2 = 2², which simplifies to x+2 = 4. Solving for x, we find x = 2. So the x-intercept is (2, 0).
– To find the y-intercept, substitute x = 0 into the function: f(0) = log₂(0+2) – 2 = log₂(2) – 2 = 1 – 2 = -1. The y-intercept is, therefore, (0, -1).

In summary, the function f(x) = log₂(x+2) – 2 is a logarithmic function with a domain of x > -2, a range of all real numbers, a vertical asymptote at x = -2, an x-intercept at (2, 0), and a y-intercept at (0, -1).

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