<\/span><\/h2>\nTo find the minimum value of a piecewise function, we need to consider the different intervals or regions defined by the function and evaluate the function within each interval. Then, we compare the values obtained to determine the smallest value.<\/p>\n
Let’s illustrate this process with a simple piecewise function example:<\/p>\n
\nf(x) =
\n -2x + 1 if x \u2264 3
\n 4 if x > 3
\n<\/pre>\nTo find the minimum value of this piecewise function, we need to evaluate it for all the different regions of its domain. In this case, we have two regions: one for x \u2264 3 and one for x > 3.<\/p>\n
For x \u2264 3:
\nSubstitute the value of x into the formula for this region:
\nf(x) = -2x + 1<\/p>\n
Let’s evaluate the function for x = 2:
\nf(2) = -2(2) + 1
\n = -4 + 1
\n = -3<\/p>\n
For x > 3:
\nSubstitute the value of x into the formula for this region:
\nf(x) = 4<\/p>\n
Since this region does not depend on the value of x, the function will always yield f(x) = 4 when x is greater than 3.<\/p>\n
Comparing the values obtained, we see that f(2) = -3 is smaller than f(x) = 4 for any x > 3. Therefore, the minimum value of this piecewise function is -3.<\/p>\n