Problem 70 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(67 - 70\) , use the following function. \(f ( x ) = \left\\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.\) Multiple Choice What is the value of \(f ( 1 ) ?\) (A) 5\(/ 2 \quad\) (B) 3\(/ 2 \quad\) (C) (D) 0 1(E) does not exist

Step-by-Step Solution

Answer

The value of \(f(1)\) is 1.

Step 1: Determine which part of the piecewise function to use

The first step is to see where the value \(x = 1\) falls in the given intervals of the piecewise function. There are two intervals in the function: \(x \leq 1\) and \(x > 1\). Since \(x = 1\) is not greater than 1, we have to use the first interval \(x \leq 1\) where the function is defined as \(2 - x\).

Step 2: Substitute \(x = 1\) into the correct function interval

Now that we know that we need to use the first function interval for \(x \leq 1\), we can substitute \(x = 1\) into this part of the function. The expression \(\(2 - 1\) \) is used to find \(f(1)\).

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

Substituting \(x = 1\) into the expression gives us \(2-1=1\). Therefore, \(f(1) = 1\).

Key Concepts

Function Evaluation

Function evaluation is all about finding the output of a function when you input a specific value for the variable. In this case, you have a piecewise function, which means the function is defined by different expressions depending on the value of the input. To evaluate a piecewise function like \(f(x)\) at \(x = 1\), you must first determine which part of the function's definition to use.

Consider this: a function's definition might change depending on different input ranges, called intervals. These intervals dictate which formula to use for any given input. For \(f(x)\), there are two expressions: one for when \(x \leq 1\) and another for when \(x > 1\). Therefore, evaluating the function requires checking which interval your input falls into.

Step-by-Step Solution

Step-by-step solutions break down the process of evaluating a function into clear, manageable steps. Let's walk through the process for \(f(1)\):

1. **Identify the correct interval:** First, figure out which part of the piecewise function to use. Because \(x = 1\) and we have two intervals, \(x \leq 1\) and \(x > 1\), since 1 is equal to 1 and not greater, we use the formula for \(x \leq 1\).

2. **Substitute the value in:** With the interval determined, substitute \(x = 1\) into the expression for \(x \leq 1\), which is \(2 - x\).

3. **Compute the result:** Finally, perform the arithmetic: \(2 - 1\) which equals 1. This means \(f(1) = 1\).

These steps ensure that you consider the proper conditions and calculations to accurately determine the result.

Piecewise Intervals

Piecewise intervals describe how a function can have different formulas or equations within different domains or sections of input values. Each piece of the function corresponds to a different interval.

In our example, the piecewise function \(f(x)\) has two parts:

• \(2 - x\) for \(x \leq 1\)
• \(\frac{x}{2} + 1\) for \(x > 1\).

These intervals help break down complex functions into simpler, specific rules according to the values of the input.

If you imagine a road map, think of piecewise intervals as the different paths you might take when driving based on the directions you follow depending on where you're located. By clearly defining where each interval starts and ends, you're setting up clear rules for how to proceed, just like which turn to take at a fork in the road.