Problem 21 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Evaluate f(-2)

For \( x = -2 \), use the function rule for \( x < 0 \). This gives us \( f(x) = x^2 \). So, \( f(-2) = (-2)^2 = 4 \).

Step 2: Evaluate f(-1)

For \( x = -1 \), we again use the rule \( x^2 \) since \( x < 0 \). Thus, \( f(-1) = (-1)^2 = 1 \).

Step 3: Evaluate f(0)

Here, \( x = 0 \) and the rule for \( x \geq 0 \) applies, which is \( x + 1 \). Therefore, \( f(0) = 0 + 1 = 1 \).

Step 4: Evaluate f(1)

For \( x = 1 \), use the rule \( x + 1 \) since \( x \geq 0 \). Thus, \( f(1) = 1 + 1 = 2 \).

Step 5: Evaluate f(2)

At \( x = 2 \), apply the rule \( x + 1 \) as \( x \geq 0 \). Therefore, \( f(2) = 2 + 1 = 3 \).

Key Concepts

Function Evaluation

Function evaluation essentially means finding the value of a function at a specific point. It is like plugging numbers into an equation to see what output you get. For example, if you have a function \( f(x)=x^2 \), and you want to evaluate it at \( x=3 \), you replace \( x \) with \( 3 \) and compute the answer, which in this case would be \( 9 \).

In piecewise functions, you need to pay attention to conditions associated with different rules to evaluate the function properly. Each interval or condition will have its own formula, so selecting the right one is crucial. As you've seen in the problem provided, determining which rule to apply based on the value of \( x \) is the first step. Follow these guidelines to execute function evaluations smoothly:

• Identify the specific function rule that corresponds to your input value based on the given conditions.
• Substitute your input value into the selected rule.
• Solve the expression to get the output for the function at that input value.

Mastering the art of function evaluation helps in finding outputs for more complex functions later in math.

Piecewise Defined Function

A piecewise defined function is a function composed of multiple sub-functions, each with its own rule that applies to different parts of the domain. This means that the function changes its rule or formula depending on which part of the input it is considering. Such a function may have a simple switch in rules at certain points or could encompass vastly different behaviors in different segments of the input.

The function presented in the original exercise is a classic example of a piecewise function:

• If \( x < 0 \), the function rule is \( f(x) = x^2 \).
• If \( x \geq 0 \), the function rule is \( f(x) = x + 1 \).

This setup allows the function to exhibit different behaviors on either side of \( x = 0 \). Understanding piecewise functions helps interpret complex real-world situations where one formula cannot capture an entire scenario. It's commonly used in fields like economics or engineering where different conditions dictate changing models.

Function Rules

Function rules are like instructions for calculating the output of a function based on the input. In a piecewise function, there are often multiple function rules, each designated for a certain interval within the domain of the function. Each rule has its own expression, and it is essential to decide which rule to use based on the value of \( x \).

In the given exercise, the function consists of two rules:

• For \( x < 0 \), use the rule \( f(x) = x^2 \), meaning you square the input \( x \).
• For \( x \geq 0 \), use the rule \( f(x) = x + 1 \), meaning you add one to the input \( x \).

Using these rules correctly translates to choosing the right formula for your input. It is like following a recipe: each condition in the domain is paired with its appropriate formula. This methodical approach ensures that the right calculations are applied, despite having different formulas that might seem confusing at first glance. Familiarizing yourself with function rules will provide clarity when facing intricate problems involving multiple potential calculations.