Problem 46 - College Algebra

Question

For the following exercises, given each function \(f,\) evaluate \(f(-3), f(-2), f(-1),\) and \(f(0)\) $$ f(x)=\left\\{\begin{array}{ccc}{x+1} & {\text { if }} & {x<-2} \\ {-2 x-3} & {\text { if }} & {x \geq-2}\end{array}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Evaluate f(-3)

Given that -3 is less than -2, use the first piece of the function: \[ f(x) = x + 1. \] Substitute -3 for \( x \), giving \[ f(-3) = -3 + 1 = -2. \] Hence, \( f(-3) = -2. \)

Step 2: Evaluate f(-2)

For \( x = -2 \), this value is exactly equal to \( -2 \), and according to the function, we use the second piece: \[ f(x) = -2x - 3. \] Substitute -2 for \( x \) to get \[ f(-2) = -2(-2) - 3 = 4 - 3 = 1. \] So, \( f(-2) = 1. \)

Step 3: Evaluate f(-1)

Since -1 is greater than -2, we use the second piece of the function: \[ f(x) = -2x - 3. \] Substitute -1 for \( x \), resulting in \[ f(-1) = -2(-1) - 3 = 2 - 3 = -1. \] Therefore, \( f(-1) = -1. \)

Step 4: Evaluate f(0)

Here, 0 is greater than -2, so we again use the second piece of the function: \[ f(x) = -2x - 3. \] Substitute 0 for \( x \), and we have \[ f(0) = -2(0) - 3 = 0 - 3 = -3. \] Thus, \( f(0) = -3. \)

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics that involves finding the output value of a function for a given input. Imagine a function as a machine that processes an input to produce an output. The rule by which the input is processed can often be represented by a formula. To evaluate a function, you substitute the given input value into the formula to find the corresponding output.
For example, if we have a function defined as \( f(x) \) with different expressions for different ranges of \( x \), we substitute the specific value of \( x \) into the appropriate piece of the function, determined by its conditions.

• Evaluate the expression by substituting the input value.
• Apply any arithmetic operations according to the rule of the function.
• Find the precise output value corresponding to your input.

Emphasizing function evaluation ensures that students can systematically determine outputs, which is crucial for understanding more complex mathematical concepts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are the building blocks of algebra and critical to constructing mathematical models. These expressions follow specific rules related to operations like addition, subtraction, multiplication, and division.
In the context of a piecewise function, each piece is an algebraic expression that represents a specific rule that applies within a particular interval of the variable.
For instance, consider the piecewise function:

• \( f(x) = x + 1 \) for \( x < -2 \)
• \( f(x) = -2x - 3 \) for \( x \geq -2 \)

Each algebraic expression must be simplified correctly when evaluating, ensuring the correct operations are performed in the right order. The understanding of algebraic expressions provides the basis for evaluating piecewise functions and more complex mathematical problems.

Conditional Statements

Conditional statements in mathematics are used to define situations where certain conditions must be met before applying specific rules or formulas. They are crucial in defining piecewise functions, which involve different expressions applied depending on the input value intervals.
Piecewise functions use conditional statements to switch between different algebraic expressions based on whether an input value belongs to certain predefined intervals.
For example, consider the function:

• Use \( f(x) = x + 1 \) if \( x < -2 \)
• Use \( f(x) = -2x - 3 \) if \( x \geq -2 \)

These statements are explicit conditions that determine the piece of the function to be used. Understanding how to apply these conditions allows students to accurately evaluate and graph complex functions and enhance their problem-solving skills in mathematical modeling.