Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function at each specified value of the independent variable and simplify. $$f(x)=\left\\{\begin{array}{ll}x^{2}-4, & x \leq 0 \\ 1-2 x^{2}, & x>0\end{array}\right.$$ (a) \(f(-2)\) (b) \(f(0)\) (c) \(f(1)\)

Step-by-Step Solution

Answer

The values of \(f(-2)\), \(f(0)\), and \(f(1)\) are 0, -4, and -1, respectively.

Step 1: Evaluate \(f(-2)\)

Since -2 is less than or equal to zero, we apply the function \(x^{2}-4\). So, \(f(-2) = (-2)^{2} - 4 = 4 - 4 = 0.

Step 2: Evaluate \(f(0)\)

Since 0 is equal to zero, we apply the function \(x^{2}-4\). So, \(f(0) = (0)^{2} - 4 = 0 - 4 = -4.

Step 3: Evaluate \(f(1)\)

Since 1 is greater than zero, we apply the function \(1-2x^{2}\). So, \(f(1) = 1 - 2*(1)^{2} = 1 - 2 = -1.

Key Concepts

Function Evaluation

Evaluating a piecewise function consists of determining which condition applies based on a given value for the independent variable, and then applying the corresponding rule to calculate the function's value.
For example, in our function, we have two different rules:

• If the value of the independent variable, denoted by \(x\), is less than or equal to zero, we use the rule \(x^2 - 4\).
• If it is greater than zero, we switch to using \(1 - 2x^2\).

Let's take a look at how we evaluate the function using these rules. When \(x = -2\), since \(-2\) is less than 0, we use the first rule: \((-2)^2 - 4 = 4 - 4 = 0\).
For \(x = 0\), we again use the first rule because 0 is less than or equal to 0: \(0^2 - 4 = -4\).
Lastly, for \(x = 1\), we apply the second rule as 1 is greater than 0: \(1 - 2 imes 1^2 = 1 - 2 = -1\). Each evaluation follows a simple process of applying the right expression based on value conditions.

Independent Variables

The independent variable is a key component in any function. It serves as the input for computations, and the value you substitute into the function to find the result.
In our piecewise function, the independent variable is denoted as \(x\). The function's value changes depending on the value of \(x\).
Different values of \(x\) allow us to use different parts of our piecewise function. It's essential to verify whether each specific \(x\) value satisfies the condition linked to its respective part.

• With our current function structure, one part uses \(x\) values where \(x \leq 0\).
• The other part covers \(x > 0\).

Understanding which condition \(x\) satisfies is critical. It determines which mathematical operation to apply to accurately evaluate the function's outcome. Always structure your checks in sequence: identify if \(x\) belongs to one condition, before moving to the next.

Function Simplification

Simplification is the final step after evaluation and involves reducing expressions for clarity without changing their value. This process is vital for ensuring that your final answer makes sense and is easy to interpret.
For instance, after plugging the value of \(x\) into the formula, arithmetic operations are performed to reach a simpler form. In a case where \(f(-2)\) was evaluated using \(x^2 - 4\), it resulted in \(4 - 4\), which simplifies neatly down to \(0\).
Similarly, simplifying \(f(0)\) results in \(0^2 - 4 = -4\), and \(f(1)\) becomes \(1 - 2(1)^2 = 1 - 2 = -1\). These steps entail not only substituting values but also performing basic arithmetic to ensure the result is as simple as possible. In essence, simplification is about making the function's output easily comprehensible, while retaining all necessary numerical components.