Analyzing the function f(x) = √(|x-2|) | True statement and graph analysis

Let f be the function defined by f(x) = sqrt(|x-2|) for all x. Which of the following statements is true?

To determine which statement is true, let’s analyze the given function f(x) = √(|x-2|)

To determine which statement is true, let’s analyze the given function f(x) = √(|x-2|).

The function f(x) involves the square root of the absolute value of x minus 2.

1. f(x) is defined for all real values of x.
This statement is true. The function f(x) is defined for all real numbers because there are no restrictions on the domain of x.

2. The graph of f(x) is always increasing.
This statement is not true. The graph of f(x) is not always increasing; it depends on the value of x. For x ≤ 2, the graph decreases because the absolute value of (x-2) is negative, resulting in imaginary values. However, for x > 2, the graph increases because the absolute value of (x-2) is positive, resulting in real values.

3. The graph of f(x) has a vertical asymptote at x = 2.
This statement is not true. Since f(x) is defined for all real numbers, there is no vertical asymptote at x = 2. The function is continuous at x = 2.

4. The y-intercept of the graph of f(x) is -2.
This statement is not true. To find the y-intercept, we set x = 0 and evaluate the function: f(0) = √(|0-2|) = √2. Therefore, the y-intercept is √2, not -2.

In conclusion, the only true statement is:
– The function f(x) is defined for all real values of x.

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