why is f(x)=|x-5| not linear ??? and how can one figure out that it's not linear ???
To determine if a function is linear, we need to check whether it satisfies the properties of linearity, which are:
1. The function must be defined for all real numbers.
2. The function must have a constant rate of change (slope) throughout its domain.
3. The graph of the function must be a straight line.
Let's analyze the function f(x) = |x - 5| using these properties:
1. The function f(x) = |x - 5| is defined for all real numbers. So, it satisfies the first property.
2. To determine if the function has a constant rate of change, we need to analyze its slope. In the case of a linear function, the slope is the same for any two points on its graph. Let's take two points: (3, |3 - 5|) = (3, 2) and (7, |7 - 5|) = (7, 2). Calculating the slope, which is the change in y divided by the change in x, we get:
Slope = (2 - 2) / (7 - 3) = 0 / 4 = 0.
Since the slope is 0 for any two points on the graph of the function, it satisfies the second property.
3. Finally, we need to examine the graph of the function. The graph of f(x) = |x - 5| is a V-shape with the vertex at (5, 0). It is not a straight line but a piecewise function composed of two linear segments. Therefore, it does not satisfy the third property, and we can conclude that f(x) = |x - 5| is not a linear function.
In summary, to determine if a function is linear, we need to examine its properties, which include checking that it has a constant rate of change and that its graph is a straight line.