Problem 57 - College Algebra with Modeling and Visualization

Question

Use f(x) to determine verbal, graphical and numerical representations. For the numerical representation use a table wish \(x=-2,-1,0,1,2\) Evaluate \(f(2).\) $$ f(x)=5-x $$

Step-by-Step Solution

Answer

\(f(2) = 3\)

Step 1: Understand the function

The given function is \(f(x) = 5 - x\). This defines a linear function where the slope is -1 and the y-intercept is 5. It is important to understand that for any input \(x\), the function subtracts \(x\) from 5 to find the output \(f(x)\).

Step 2: Create a numerical table

To represent the function numerically, we create a table with the provided \(x\) values: -2, -1, 0, 1, 2. Compute the corresponding \(f(x)\) for each value. \[\begin{array}{c|c}x & f(x) \\hline-2 & 5 - (-2) = 5 + 2 = 7 \-1 & 5 - (-1) = 5 + 1 = 6 \0 & 5 - 0 = 5 \1 & 5 - 1 = 4 \2 & 5 - 2 = 3 \\end{array}\]

Step 3: Graphical representation

Plot the points \((-2,7)\), \((-1,6)\), \((0,5)\), \((1,4)\), and \((2,3)\) on a coordinate plane. Connect the points with a straight line. This line represents the function graphically, showing a decrease in \(f(x)\) as \(x\) increases, consistent with a slope of -1.

Step 4: Verbal representation

The function \(f(x) = 5 - x\) can be described verbally as: "The value of the function is the number 5 decreased by the input \(x\)." As the input \(x\) increases, \(f(x)\) decreases linearly.

Step 5: Evaluate \(f(2)\)

Substitute \(x = 2\) into the function to find \(f(2)\): \[ f(2) = 5 - 2 = 3 \] Thus, \(f(2) = 3\).

Key Concepts

Numerical Representation

In this part, we aim to understand how to represent the linear function numerically. The function given is \( f(x) = 5 - x \), a simple linear equation. The process involves substituting different values of \( x \) into the equation to calculate \( f(x) \).

Here's how it works: for each \( x \) value in the set \( \{-2, -1, 0, 1, 2\} \), you will compute \( f(x) \) by plugging \( x \) into the function. For example:

• For \( x = -2 \), \( f(-2) = 5 - (-2) = 7 \)
• For \( x = -1 \), \( f(-1) = 5 - (-1) = 6 \)
• For \( x = 0 \), \( f(0) = 5 - 0 = 5 \)
• For \( x = 1 \), \( f(1) = 5 - 1 = 4 \)
• For \( x = 2 \), \( f(2) = 5 - 2 = 3 \)

By performing these calculations, we capture the numerical behavior of the function and can summarize it in a table format. This makes it easier to see how the function behaves across a range of values.

Graphical Representation

Moving on to graphical representation, we convert the numerical representation into a visual one. This helps us understand the function's behavior at a glance. Using the previous calculations, we plot the points \((-2, 7)\), \((-1, 6)\), \((0, 5)\), \((1, 4)\), and \((2, 3)\) on a Cartesian coordinate plane.

To graph these, set up your graph with \( x \)-axis and \( f(x) \)-axis marked. Locate each calculated point and mark it on the graph.

• Start with \((-2, 7)\), two units left from the origin and seven units up.
• Next, go to \((-1, 6)\), left one unit from zero and move up six.
• Place \((0, 5)\) right at five on the \( f(x) \)-(vertical) axis.
• Mark \((1, 4)\), moving right one and up four.
• Finally, plot \((2, 3)\), two units right from the origin and three up.

Connect them in a straight line which visually shows the function's slope of \(-1\). As \( x \) increases, \( f(x) \) decreases, confirming the linear relationship visually.

Function Evaluation

Function evaluation means substituting a specific \( x \) value into the function to determine \( f(x) \). It's a straightforward process where each step is clear and logical. For example, to evaluate \( f(2) \), we substitute \( 2 \) for \( x \) in our function \( f(x) = 5 - x \).

Insert the value:

• Replace \( x \) with \( 2 \): \( f(2) = 5 - 2 \)
• Perform the subtraction: \( f(2) = 3 \)

By evaluating the function at a specific point like \( x = 2 \), it's possible to determine the exact output, \( f(2) = 3 \), which provides a precise answer. Function evaluation is critical in verifying values and understanding function output at particular inputs.