Problem 28 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

f(-3) = 5, f(0) = 5, f(2) = 5, f(3) = 3, f(5) = 7.

Step 1: Evaluate f(-3)

Since \( -3 \leq 2 \), we use the first part of the piecewise function: \( f(x) = 5 \). Therefore, \( f(-3) = 5 \).

Step 2: Evaluate f(0)

Since \( 0 \leq 2 \), we again use the first part of the piecewise function: \( f(x) = 5 \). Therefore, \( f(0) = 5 \).

Step 3: Evaluate f(2)

Since \( 2 \leq 2 \), we still use the first part of the piecewise function: \( f(x) = 5 \). Therefore, \( f(2) = 5 \).

Step 4: Evaluate f(3)

Since \( 3 > 2 \), we use the second part of the piecewise function: \( f(x) = 2x - 3 \). Calculate \( f(3) = 2(3) - 3 = 6 - 3 = 3 \).

Step 5: Evaluate f(5)

Since \( 5 > 2 \), we use the second part of the piecewise function: \( f(x) = 2x - 3 \). Calculate \( f(5) = 2(5) - 3 = 10 - 3 = 7 \).

Key Concepts

Function Evaluation

When we talk about function evaluation in mathematics, we mean finding the output value of a function for a specific input. In this exercise, the function is defined differently for different ranges of the input value, which makes it a "piecewise function." Evaluating a piecewise function properly requires us to first determine which segment of the function applies to our specific input value.
Understanding the piecewise function given in the exercise will help clarify this idea:

• If the input \( x \) is 2 or less, use the constant function \( f(x) = 5 \).
• If \( x \) is greater than 2, use the linear function \( f(x) = 2x - 3 \).

Now, by applying these rules, the evaluation process becomes a straightforward task of deciding which rule to use and then substituting in the value of \( x \). For instance, since \( -3 \) is less than or equal to 2, the output is simply 5 according to the piecewise conditions, and thus \( f(-3) = 5 \).

Algebra

Algebra is a fascinating area of mathematics that involves working with symbols and numbers to solve problems. In this exercise, algebra is used especially when handling parts of the piecewise function that are not constants, like \( f(x) = 2x - 3 \).When evaluating an expression such as \( f(x) = 2x - 3 \), you'll often:

• Substitute the given value of \( x \) into the equation.
• Simplify the expression to get the result.

For example, to evaluate \( f(3) \), we substitute 3 into the equation: \( f(3) = 2(3) - 3 = 6 - 3 = 3 \). This shows one way algebra helps in calculating outcomes within a given set of rules. Algebra allows us to describe general patterns by using equations, letting us tackle various problems by following structured manipulations of those algebraic expressions.

Mathematical Notation

Mathematical notation is crucial for expressing ideas clearly and concisely. In this exercise, you encounter several types of notation which are essential for understanding piecewise functions.Here are some key elements of notation from this problem:

• The use of curly brackets \( \{ \} \) in defining the piecewise function shows that it consists of multiple parts under different conditions.
• The inequality symbols \( \leq \) and \( > \) help us determine which part of the function to apply.

Understanding these symbols is important for interpreting the conditions under which parts of a piecewise function operate. This kind of function may initially seem complex, but correct use of mathematical notation helps simplify and lay out the steps needed to evaluate the function for different inputs clearly. Knowing this notation is key to working efficiently in algebra, enabling you to immediately perceive the structure and rules that dictate how functions must be evaluated.