Problem 46 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Evaluate f(-3)

Since \(-3 < -2\), we use the first part of the piecewise function, \(f(x) = x + 1\). Thus,\(f(-3) = -3 + 1 = -2\).

Step 2: Evaluate f(-2)

Because \(-2 \geq -2\), we use the second part of the piecewise function, \(f(x) = -2x - 3\). Calculating this gives us:\(f(-2) = -2(-2) - 3 = 4 - 3 = 1\).

Step 3: Evaluate f(-1)

As \(-1 > -2\), we apply the second part of the function, \(f(x) = -2x - 3\). Thus,\(f(-1) = -2(-1) - 3 = 2 - 3 = -1\).

Step 4: Evaluate f(0)

Since \(0 > -2\), we continue using the second part of the function,\(-2x - 3\). Thus,\(f(0) = -2(0) - 3 = 0 - 3 = -3\).

Key Concepts

Function Evaluation

In mathematics, function evaluation is the process of finding the output of a function given a specific input. This is a fundamental concept as it allows us to understand how a function behaves based on its formula or definition. When dealing with piecewise functions, the function evaluation involves determining which piece of the function applies, depending on the input value.
For example, we are given a piecewise function:

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

To evaluate this function at points like -3, -2, -1, and0, we must check which condition each input satisfies. This will tell us which rule to apply. Evaluating \( f(-3) \) would use the first piece because \(-3 < -2\). Meanwhile, inputs like -2, -1, and0 would all use the second piece, since they are greater than or equal to -2.

Step-by-step Solutions

Solving mathematical problems step by step is a crucial skill. It ensures a thorough understanding of methods and processes. Let's look at how to break down the evaluation of each function point in detail using step-by-step solutions.

• **Step 1**: Evaluate \( f(-3) \)
As \(-3 < -2\), we apply the first piece: \( f(x) = x + 1 \).
This results in \( f(-3) = -3 + 1 = -2 \).

• **Step 2**: Evaluate \( f(-2) \)
Because \(-2 \geq -2\), use the second piece: \( f(x) = -2x - 3 \).
Thus, \( f(-2) = -2(-2) - 3 = 4 - 3 = 1 \).

• **Step 3**: Evaluate \( f(-1) \)
With \(-1 > -2\), we continue with the second rule: \( f(x) = -2x - 3 \).
Calculating gives us \( f(-1) = -2(-1) - 3 = 2 - 3 = -1 \).

• **Step 4**: Evaluate \( f(0) \)
Since \( 0 > -2 \), we again use the second piece: \( -2x - 3 \).
This results in \( f(0) = -2(0) - 3 = -3 \).

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In this context, it's essential to handle variables and expressions accurately. The problem at hand involves piecewise functions, which are a common topic in algebra.
When solving or evaluating piecewise functions like the one given, understanding algebraic operations such as add, subtract, multiply, and apply conditions based on inequalities is key. The piecewise function has conditions attached to each piece, and being comfortable with inequalities—like determining whether an input is less than or greater than a given value—guides which part of the function to use.
For instance:

• The "if \( x < -2 \)" condition signifies the set of inputs for that function piece.
• "\( x \geq -2 \)" indicates another condition.

By following these, algebra helps us perform operations correctly and find the function values at different inputs.