answer:
The given function f(x) is defined piecewise as follows:
f(x) = x, for 0 < x ≤ 2
f(x) = x² ln 2, for x = 2
Let's break down the function and analyze each part:
1. For values of x between 0 and 2 (excluding 0), the function f(x) is defined as x. This means that for any value of x within this range, the output of the function will be equal to the input x. For example, if we substitute x = 1 into the function, f(1) = 1.
2. At x = 2, the function takes a different form. The function f(x) is defined as x² ln 2 when x = 2. This means that when x equals 2, the output of the function will be the square of 2 multiplied by the natural logarithm of 2. In this case, if we substitute x = 2 into the function, f(2) = 2² ln 2 = 4 ln 2.
To summarize:
- For values of x between 0 and 2 (excluding 0), the function f(x) equals x.
- At x = 2, the function f(x) equals 4 ln 2.
It's important to note that the function f(x) is continuous at x = 2 since the limit of the function as x approaches 2 from the left (x < 2) is 2, and the value of the function at x = 2 is also 2. Therefore, there are no discontinuities or jumps in the function at x = 2.
so real