Problem 24 - Algebra and Trigonometry

Question

Evaluate the piece wise defined function at the indicated values. $$ \begin{array}{ll}{f(x)=\left\\{\begin{array}{ll}{3 x} & {\text { if } x<0} \\\ {x+1} & {\text { if } 0 \leq x \leq 2} \\ {(x-2)^{2}} & {\text { if } x>2}\end{array}\right.} \\ {f(-5), f(0), f(1), f(2), f(5)}\end{array} $$

Step-by-Step Solution

Answer

\( f(-5) = -15, f(0) = 1, f(1) = 2, f(2) = 3, f(5) = 9 \).

Step 1: Evaluate f(-5)

Since \( x = -5 \) is less than 0, use the rule \( f(x) = 3x \). Substitute \( x = -5 \) into the function: \( f(-5) = 3(-5) = -15 \). So, \( f(-5) = -15 \).

Step 2: Evaluate f(0)

Since \( x = 0 \) falls within the interval \( 0 \leq x \leq 2 \), use the rule \( f(x) = x + 1 \). Substitute \( x = 0 \) into the function: \( f(0) = 0 + 1 = 1 \). Thus, \( f(0) = 1 \).

Step 3: Evaluate f(1)

\( x = 1 \) is also within the interval \( 0 \leq x \leq 2 \), so we again use the rule \( f(x) = x + 1 \). Substitute \( x = 1 \): \( f(1) = 1 + 1 = 2 \). Hence, \( f(1) = 2 \).

Step 4: Evaluate f(2)

\( x = 2 \) is at the upper end of the interval \( 0 \leq x \leq 2 \), so we use \( f(x) = x + 1 \). Substitute \( x = 2 \): \( f(2) = 2 + 1 = 3 \). Therefore, \( f(2) = 3 \).

Step 5: Evaluate f(5)

Since \( x = 5 \) is greater than 2, use the rule \( f(x) = (x-2)^2 \). Substitute \( x = 5 \): \( f(5) = (5-2)^2 = 3^2 = 9 \). Hence, \( f(5) = 9 \).

Key Concepts

Function Evaluation

When approaching the evaluation of a function, especially a piecewise one, it is essential first to understand how to determine which part of the function to use. A piecewise function is defined by multiple sub-functions, each applicable to a specific interval of the domain of the main function.
The critical first step is identifying which condition or interval the input value satisfies. Let's take the function as given:- If the input value (say, \( x \)) is less than 0, the function is represented by \( f(x) = 3x \).- If \( x \) lies between 0 and 2 inclusive, \( f(x) = x + 1 \).- If \( x \) is greater than 2, the function follows \( f(x) = (x-2)^2 \).After identifying the correct segment of the piecewise function, you simply substitute your input value into that specific function to get your result. This simple two-step process of identification and substitution makes tackling piecewise functions straightforward and methodical.

Interval Notation

Interval notation is an efficient way to represent the set of numbers between any two endpoints. It is frequently used in mathematics to describe the domains of piecewise functions. The notation is characterized by using brackets and parentheses:- Square brackets \([ \text{ and } ]\) represent endpoints that are included, also known as closed intervals.- Parentheses \(( \text{ and } )\) denote endpoints that are not included, termed open intervals.For our piecewise function, observe:- \( x < 0 \) is represented by the interval \( (-\infty, 0) \) with a parenthesis at 0, indicating "less than."- \( 0 \leq x \leq 2 \) is denoted as \([0, 2]\), using square brackets to include 0 and 2 as part of the interval.- \( x > 2 \) is written as \((2, \infty)\), with a parenthesis at 2 showing it is not included in this part of the piecewise function.Using interval notation helps clearly communicate which parts of the function apply to different sets of numbers, making mathematical interpretations precise and straightforward.

Substitution Method

The substitution method in mathematics is a straightforward technique where you replace a variable within a function with a specific value or expression. This method is particularly useful when dealing with functions, including piecewise ones.When evaluating piecewise functions with substitution, follow these steps:

• Select the correct portion of the function by identifying which interval the input falls within.
• Replace the variable in the function with the given number.

For example, if you're tasked with finding \( f(0) \) in our piecewise-defined function, you first determine that 0 lies in the interval \([0, 2]\). Therefore, select the function segment \( f(x) = x + 1 \) and substitute \( x \) for 0, simply calculating \( 0 + 1 \), which equals 1.The substitution method is a highly efficient approach for evaluating any function value, especially when input values are given and you might deal with different rules based on the input.