Problem 51 - College Algebra

Question

For the following exercises, given each function \(f\) , evaluate \(f(-1), f(0), f(2),\) and \(f(4)\) $$ f(x)=\left\\{\begin{array}{ll}{5 x} & {\text { if } \quad x<0} \\ {3} & {\text { if } \quad 0 \leq x \leq 3} \\ {x^{2}} & {\text { if }} & {x>3}\end{array}\right. $$

Step-by-Step Solution

Answer

\(f(-1) = -5\), \(f(0) = 3\), \(f(2) = 3\), \(f(4) = 16\).

Step 1: Identify the Function Definition

We are given a piecewise-defined function. It has three parts: 1. If \(x < 0\), then \(f(x) = 5x\).2. If \(0 \leq x \leq 3\), then \(f(x) = 3\).3. If \(x > 3\), then \(f(x) = x^2\).We will use this information to evaluate the function at the specified points: \(-1, 0, 2,\) and \(4\).

Step 2: Evaluate at x = -1

Since \(x = -1\) is less than 0, we use the first part of the function: \(f(x) = 5x\).Substitute \(-1\) into the function:\[f(-1) = 5(-1) = -5\]

Step 3: Evaluate at x = 0

Since \(0 \leq x \leq 3\), according to the function definition, we use the second part: \(f(x) = 3\).Thus,\[f(0) = 3\]

Step 4: Evaluate at x = 2

Again, since \(0 \leq 2 \leq 3\), we use the second part of the function: \(f(x) = 3\).Thus,\[f(2) = 3\]

Step 5: Evaluate at x = 4

Since \(x = 4\) is greater than 3, we use the third part of the function: \(f(x) = x^2\).Substitute \(4\) into the function:\[f(4) = 4^2 = 16\]

Key Concepts

Function Evaluation

Function evaluation is about determining the output of a function for a specific input. In our example, we have a piecewise function, meaning it uses different rules based on the value of the input variable, in this case, \(x\). Each piece or part of the function applies to specific ranges of \(x\). The concept of function evaluation for piecewise functions involves following these steps:

• Identify the correct rule that applies to the given input value \(x\).
• Substitute the value into the corresponding rule to compute the result.

For instance, when \(x = -1\), we check each piece of the function to see which condition \(-1\) fits into. Since \(-1\) is less than 0, we apply the rule \(f(x) = 5x\). After substituting \(x = -1\), we compute \(f(-1) = 5(-1) = -5\). Evaluating each piece of the function this way helps us understand how different conditions affect the output of the function.

Algebraic Functions

Algebraic functions are mathematical expressions made up of variables and constants using algebraic operations such as addition, subtraction, multiplication, division, and powers. These functions form the basis of many mathematical concepts.

In the given piecewise function, each piece is an algebraic expression. Let's explore each:

• The first piece is given by \(f(x) = 5x\). It’s a linear expression representing a line with slope 5, applicable when \(x < 0\).
• The second piece is a constant function: \(f(x) = 3\), valid for the interval \(0 \leq x \leq 3\). No matter the input within this range, the output remains constant.
• The third part is a quadratic expression: \(f(x) = x^2\), used when \(x > 3\). This represents a parabolic curve opening upwards.

Understanding the nature of these algebraic functions helps us see how input values are transformed into outputs based on their properties.

Mathematical Notation

Mathematical notation is the system of symbols used to express mathematical ideas and relationships. It's essential for conveying complex concepts clearly and concisely. In piecewise functions, notation helps specify rules for different intervals of the function domain clearly.

• The brace \(\{\) beside the function indicates that multiple rules define it, a hallmark of piecewise-defined functions.
• Each condition like \(x < 0\) establishes the intervals over which each rule applies.
• Notations like \(0 \leq x \leq 3\) specify ranges in an inclusive manner, meaning all values from 0 to 3, including the endpoints, are included.
• Expressions such as \(x^2\) use standard mathematical symbols for operations like squaring, making the rules intuitive once you know the symbols.

By mastering this notation, you can accurately interpret and apply functions, making your work in algebra more precise and efficient.