Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function at each specified value of the independent variable and simplify. $$f(x)=\left\\{\begin{array}{ll}5-2 x, & x<0 \\ 5, & 0 \leq x<1 \\ 4 x+1, & x \geq 1\end{array}\right.$$ (a) \(f(-4)\) (b) \(f(0)\) (c) \(f(1)\)

Step-by-Step Solution

Answer

The evaluated results are \(f(-4) = 13\), \(f(0) = 5\) and \(f(1) = 5\).

Step 1: Determine the appropriate expression for \(f(-4)\)

For \(f(-4)\), find the interval that contains -4 in the piecewise function. In this case, -4 is less than 0, so we will use the expression \(5-2x\).

Step 2: Evaluate \(f(-4)\)

Substitute -4 into \(5-2x\) to get \(f(-4)\). This gives \(5 - 2(-4) = 13.\)

Step 3: Determine the appropriate expression for \(f(0)\)

For \(f(0)\), determine which interval contains 0. Here, \(0 \geq 0\) and \(0 < 1\), so we use the expression \(5\). As there are no variables to substitute, \(f(0) = 5\).

Step 4: Determine the appropriate expression for \(f(1)\)

For \(f(1)\), determine which interval contains 1. In this case, \(1 \geq 1\), so we use the expression \(4x+1\).

Step 5: Evaluate \(f(1)\)

Substitute 1 into \(4x+1\) to get \(f(1)\), thus giving \(4(1) + 1 = 5\).

Key Concepts

Evaluating Functions

When you evaluate a function, you're essentially determining the output of the function for a particular input. This is like asking a function, "What is your value when I plug in this number?" For example, in the context of a piecewise function, evaluating means identifying which piece or expression of the function to use based on the given input value.

In our example, we have the function defined as multiple expressions based on the value of \(x\). To evaluate it for a specific \(x\), we need to:

• Identify the expression that corresponds to the specific interval the number \(x\) falls into.
• Substitute \(x\) into that chosen expression.
• Perform any arithmetic operations to simplify if needed.

Understanding how to evaluate functions is essential, especially when dealing with complex functions defined by more than one formula.

Piecewise Expressions

A piecewise function is a function composed of multiple sub-functions, each applying to a certain interval of the main function's domain. Think of it like a menu with different meals available only at certain times of the day.

For instance, the function \(f(x)\) given in the exercise is a classic example. It defines one expression for \(x < 0\), another for \(0 \leq x < 1\), and yet another for \(x \geq 1\). This allows the function to exhibit different behaviors depending on the input value.

By understanding how piecewise expressions work, you can correctly evaluate and graph complex functions that can't be expressed with a single formula. It also demonstrates the real-world applications of functions where different conditions lead to different outcomes.

Function Intervals

Function intervals are the backbone of understanding piecewise functions. Each part of a piecewise function corresponds to a specific interval, which is a range of numbers for which a particular expression is valid.

To determine which expression to use when evaluating a function, you must first understand the intervals. Here's how you can break it down:

• Look at each condition or inequality attached to the expressions. These define the boundaries of each interval.
• Decide which interval your input value falls into based on these conditions.
• Use the expression associated with that interval.

In the exercise, for \(f(-4)\), \(f(0)\), and \(f(1)\), the intervals guide which piece of the function to use. This systematic approach ensures you get the right results, making the concept of intervals fundamental when handling piecewise functions.