Problem 52 - Precalculus with Limits

Question

In Exercises 37-52, evaluate the function at each specified value of the independent variable and simplify. \( f(x) = \left\\{ \begin{array}{ll} 4 - 5x, & \mbox{ \) x \le -2 \(} \\ 0, & \mbox{ \) -2 < x < 2 \(} \\ x^2+1, & \mbox{ \) x \ge 2 \(} \end{array} \right.\) (a) \(f(-3)\) (b) \(f(4)\) (c) \(f(-1)\)

Step-by-Step Solution

Answer

The evaluated function values are: (a) 19, (b) 17, (c) 0

Step 1: Determine the Correct Sub-Function for Each Given x

(a) For \(f(-3)\), because -3 is less than -2\n (b) For \(f(4)\), because 4 is greater than or equal to 2\n (c) For \(f(-1)\), because -1 lies between -2 and 2

Step 2: Substitute the value of x into the Sub-Function

(a) Substitution gives: \(f(-3) = 4 - 5(-3) = 4 + 15 = 19\)\n (b) Substitution gives: \(f(4) = 4^2 + 1 = 16 + 1 = 17\)\n (c) Substitution gives: \(f(-1) = 0\) as the value of function is zero for any x that lies between -2 and 2

Step 3: Conclusion

The result for (a) \(f(-3)\) is 19, for (b) \(f(4)\) is 17, and for (c) \(f(-1)\) is 0.

Key Concepts

Function Evaluation

When dealing with piecewise functions, a crucial step is correctly evaluating them at specific values. A piecewise function is composed of multiple sub-functions, each defined over a certain interval. To evaluate such a function:

• Identify which part of the piecewise function to use. Each sub-function is paired with a condition, often expressed as an inequality (like \( x \leq -2 \) or \( x \geq 2 \)).
• Apply the appropriate sub-function by substituting the given value of the independent variable (such as \( x \)) into its formula.
• Perform the necessary arithmetic calculations to find the result, often simplifying the expression if needed.

Breaking down the function evaluation into these steps ensures we correctly match the value of \( x \) with its corresponding output. This process not only helps in obtaining accurate results but also reinforces understanding of how different segments of a piecewise function are applied in different scenarios.

Inequalities

Inequalities play an essential role in piecewise functions as they define the domains of the sub-functions. In a mathematical context, inequalities are expressions that determine the relationship between two variables or expressions indicating when one value is larger or smaller than another.

• Inequalities like \( x \leq -2 \) or \( x \geq 2 \) set boundaries where one sub-function ends and another begins.
• For example, if the given value is \( x = -3 \), use the inequality \( x \leq -2 \) to determine which equation to employ from the piecewise function.
• Similarly, for values that lie strictly between two mathematical expressions, like \( -2 < x < 2 \), a different sub-function is defined, resulting in different outcomes when evaluated.

Understanding how inequalities classify the range of variables helps in determining which part of a piecewise function should be applied. This systematic approach prevents errors and ensures consistency when evaluating these types of functions.

Substitution Method

Once the correct segment of a piecewise function is selected using the inequalities, the substitution method allows us to compute the value of the function accurately.

• The substitution method involves replacing the variable \( x \) in the selected sub-function with its given value.
• For instance, to find \( f(-3) \) for a piecewise function, substitute \(-3\) into the expression \(4 - 5x\) as it fits the condition \( x \leq -2 \), leading to the calculation \(4 - 5(-3) = 19\).
• This approach is straightforward: simply insert the specific \( x \) value, then compute and simplify the resulting expression to obtain the final result.
• Correct substitution is key to functioning piecewise functions effectively, making sure that each \( x \) value translates into the accurate segment of the function for evaluation.

In practice, the substitution method proves to be a dependable tool for handling piecewise functions, facilitating the transition from theoretical understanding to practical application.