Analyzing the Function f(x) = |x| + 2: Domain, Range, Intercepts, Symmetry, and Graph.

f(x) = |x| + 2

To understand and work with the function f(x) = |x| + 2, we need to know that |x| represents the absolute value of x

To understand and work with the function f(x) = |x| + 2, we need to know that |x| represents the absolute value of x. The absolute value of a number is its distance from zero on a number line, so it is always a positive value.

Now, let’s analyze the function f(x) = |x| + 2.

1. Domain: The domain of a function is the set of all possible input values (x-values). In this case, since there are no restrictions or limitations on the input, the domain of this function is all real numbers, (-∞, ∞).

2. Range: The range of a function is the set of all possible output values (y-values). In this case, we can see that the |x| term will always result in a non-negative value since the absolute value is always positive. Then, when we add 2 to that non-negative value, the resulting range will be all real numbers greater than or equal to 2, [2, ∞).

3. X-intercepts: To find the x-intercepts, we set f(x) = 0 and solve for x. In this case, since |x| is always positive, the equation |x| + 2 = 0 has no solutions. Therefore, there are no x-intercepts for this function.

4. Y-intercept: To find the y-intercept, we set x = 0 and evaluate f(x). Substituting x = 0 into the function, we get f(0) = |0| + 2 = 0 + 2 = 2. So, the y-intercept is (0, 2).

5. Symmetry: The function f(x) = |x| + 2 is not symmetric with respect to the x-axis or y-axis. It exhibits only origin symmetry or symmetry about the point (0, 2).

6. Plotting the graph: To graph the function, we can start by plotting the y-intercept (0, 2). Then, considering the behavior of |x|, we can sketch the graph as two separate lines, one for positive values of x and one for negative values of x. These lines increase at a constant rate, as the magnitude of x increases.

For positive x-values, the graph of f(x) = |x| would be a straight line with a slope of +1, starting from the y-intercept (0, 2). Then, we add 2 to the resulting points to obtain the graph of f(x) = |x| + 2.

For negative x-values, we take the absolute value of x and follow the same process, resulting in another straight line with a slope of +1. Again, we add 2 to the resulting points to obtain the graph of f(x) = |x| + 2.

Therefore, the graph of f(x) = |x| + 2 consists of two diagonal lines, intersecting at the point (0, 2), and increasing at a constant rate on both sides.

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