Understanding the Function f(x) = log₂(x-2) | Domain, Vertical Asymptote, Range, x-intercept, y-intercept, and Graph.

f(x) = log₂(x-2)

The given function is f(x) = log₂(x-2)

The given function is f(x) = log₂(x-2). Let’s break down this function step by step:

1. Domain: The domain of the given function is the set of all real numbers greater than 2. This is because the logarithm function is only defined for positive real numbers, and (x-2) needs to be positive for the function to be defined.

2. Vertical Asymptote: In this case, there is a vertical asymptote at x = 2. As x approaches 2 from the left side, f(x) goes to negative infinity. As x approaches 2 from the right side, f(x) goes to positive infinity. This is because as x gets closer to 2, (x-2) gets closer and closer to 0, and the logarithm function of 0 is undefined. Therefore, there is a vertical asymptote at x = 2.

3. Range: The range of f(x) is the set of all real numbers. This is because the logarithm function can take any real number as an input and produce a real number as an output.

4. x-intercept: To find the x-intercept of the function, we set f(x) = 0 and solve for x:
log₂(x-2) = 0
2⁰ = x-2
1 = x-2
x = 3

Therefore, the x-intercept of the function is x = 3.

5. y-intercept: To find the y-intercept of the function, we can substitute x = 0 into the equation:
f(0) = log₂(0-2)
f(0) = log₂(-2)

Here, we face an issue because the logarithm function is only defined for positive numbers, so log₂(-2) is undefined. Thus, the function does not have a y-intercept.

6. Graph: The graph of f(x) = log₂(x-2) will be a curve that approaches the vertical asymptote at x = 2. It will pass through the x-intercept at x = 3. As x increases, the function values will also increase, but at a diminishing rate.

Please note that when working with logarithmic functions, it is important to pay attention to the restrictions and limitations of the domain and the behavior around vertical asymptotes to ensure the correct interpretation and application of the function.

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