Problem 30 - Algebra and Trigonometry

Question

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

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)

For the value of \(x = -5\), we look for the condition that applies. Since \(-5 < 0\), we use the function definition \(f(x) = 3x\). Thus, \(f(-5) = 3(-5) = -15\).

Step 2: Evaluate f(0)

For \(x = 0\), we check the conditions. The piecewise function specifies that for \(0 \leq x \leq 2\), the formula is \(f(x) = x + 1\). Therefore, \(f(0) = 0 + 1 = 1\).

Step 3: Evaluate f(1)

Here \(x = 1\), and it also falls in the range \(0 \leq x \leq 2\). So the formula \(f(x) = x + 1\) is used. Thus, \(f(1) = 1 + 1 = 2\).

Step 4: Evaluate f(2)

For \(x = 2\), it is within the range \(0 \leq x \leq 2\). We use the same function formula \(f(x) = x + 1\). Thus, \(f(2) = 2 + 1 = 3\).

Step 5: Evaluate f(5)

For \(x = 5\), we use the condition \(x > 2\), where \(f(x) = (x - 2)^2\). Hence, \(f(5) = (5 - 2)^2 = 3^2 = 9\).

Key Concepts

Function Evaluation

Function evaluation is the process of determining the output of a function for a given input value. When faced with mathematical functions, understanding how to evaluate them correctly is crucial. When evaluating functions, it's necessary to be attentive to the rules or the functional expression defining them.

To evaluate functions effectively:

• Identify the input value you want to substitute into the function.
• Determine the applicable formula or rule if the function is piecewise-defined.
• Substitute the input value into the correct expression.
• Simplify the resulting expression to find the output.

Evaluation becomes straightforward once you follow these steps carefully.

Piecewise-Defined Function

A piecewise-defined function is a type of function characterized by different expressions depending on the input value. These functions are composed of "pieces," and each piece applies to a specific range of the input variable, typically denoted as "x."

Understanding Piecewise-Defined Functions:

• These functions are often represented by different expressions separated by set conditions.
• Each piece corresponds to a particular interval or inequality.
• Key aspects include determining which piece of the function to use based on the input value.

Piecewise functions might initially seem complex, but by breaking them into manageable parts, they can be easily understood and applied.

Mathematical Functions

Mathematical functions are foundational in mathematics. They establish a relationship between a set of inputs and their respective outputs. Functions can be linear, quadratic, exponential, etc., each characterized by its unique expressions.

Core understanding of mathematical functions involves:

• Recognizing the domain (valid input values) and range (potential output values).
• Understanding different forms of expressions that define these functions.
• Applying them to solve real-life problems in various fields like physics, economics, and engineering.

Attaining fluency with mathematical functions means not only being able to evaluate them but also comprehending their plots and behaviors.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, and division). In the context of functions, algebraic expressions provide a blueprint for calculating outputs for given inputs.

Key Features of Algebraic Expressions:

• Comprise constants (fixed numbers) and variables (represent numbers).
• Include operations which define how the constants and variables interact.
• Form the foundation for defining rules within mathematical and piecewise functions.

Algebraic expressions are indispensable tools in mathematics, enabling us to condense complex ideas into manageable forms and solve diverse types of problems.