A piecewise function is defined as shown.
[tex]f(x)=\left\{\begin{array}{cc} -x, & x \leq-1 \\ 1, & x=0 \\ x+1 & x \geq 1 \end{array}\right.[/tex]
What is the value of [tex]f(x)[/tex] when [tex]x=3[/tex]?
Answer (1)
Analyze the piecewise function and identify the relevant case for x = 3 .
Since 3 ≥ 1 , use the third case: f ( x ) = x + 1 .
Substitute x = 3 into the equation: f ( 3 ) = 3 + 1 .
Calculate the final value: 4 .
Explanation
Understanding the Problem We are given a piecewise function and asked to find the value of f ( x ) when x = 3 . The piecewise function is defined as: f ( x ) = { − x , x ≤ − 1 \1 , x = 0 \x + 1 x ≥ 1 .
Choosing the Correct Case Since x = 3 , we need to determine which case of the piecewise function applies. We see that x = 3 satisfies the condition x ≥ 1 . Therefore, we use the third case, which is f ( x ) = x + 1 .
Calculating the Value Now, we substitute x = 3 into the expression f ( x ) = x + 1 : f ( 3 ) = 3 + 1 = 4 .
Final Answer Therefore, the value of f ( x ) when x = 3 is 4.
Examples
Piecewise functions are used in many real-world applications, such as defining tax brackets, modeling the cost of electricity based on usage, or describing the behavior of a bouncing ball. For example, a cell phone company might charge $0.10 per minute for the first 100 minutes and $0.05 per minute for each additional minute. This can be modeled as a piecewise function, allowing you to calculate the total cost for any given number of minutes.
