Problem 54 - 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 \(f(2).\) $$ f(x)=2 x-5 $$

Step-by-Step Solution

Answer

The numerical value of \( f(2) \) is \(-1\).

Step 1: Understanding the Function

We need to represent the function \( f(x) = 2x - 5 \) verbally, graphically, and numerically. Verbally, the function states that for any input \( x \), the output \( f(x) \) is calculated by doubling the input \( x \) and then subtracting 5.

Step 2: Creating a Numerical Representation

We'll create a table of values for \( x = -2, -1, 0, 1, 2 \). Calculate \( f(x) \) for each \( x \): For \( x = -2 \), \( f(-2) = 2(-2) - 5 = -9 \) For \( x = -1 \), \( f(-1) = 2(-1) - 5 = -7 \) For \( x = 0 \), \( f(0) = 2(0) - 5 = -5 \) For \( x = 1 \), \( f(1) = 2(1) - 5 = -3 \) For \( x = 2 \), \( f(2) = 2(2) - 5 = -1 \) The table is:\( x = -2, f(x) = -9 \) \( x = -1, f(x) = -7 \) \( x = 0, f(x) = -5 \) \( x = 1, f(x) = -3 \) \( x = 2, f(x) = -1 \)

Step 3: Verbal and Graphical Representations

Verbal Representation: For every 1-unit increase in \( x \), \( f(x) \) increases by 2 since the slope is 2 and the y-intercept is -5. Graphical Representation: Plot the points calculated in the numerical representation and connect them with a line to show the graph of the linear function \( f(x) = 2x - 5 \). The line should cross the y-axis at -5 and have a positive slope of 2.

Step 4: Evaluating f(2) for Verification

As we've already calculated in our numerical table, for \( x = 2 \), \( f(2) = 2(2) - 5 = -1 \). Therefore, the value of \( f(2) \) is verified to be \(-1\).

Key Concepts

Numerical Representation

Linear functions can be represented numerically using tables that show values of the input variable, typically denoted as \( x \), and the corresponding values of the function output, \( f(x) \). For the function \( f(x) = 2x - 5 \), we create a table including various values for \( x \) and calculating their respective \( f(x) \) values. This is helpful because it allows us to easily see how changes in \( x \) affect \( f(x) \).

For example, consider \( x = -2, -1, 0, 1, \) and \( 2 \):

• When \( x = -2 \), \( f(-2) = 2(-2) - 5 = -9 \)
• When \( x = -1 \), \( f(-1) = 2(-1) - 5 = -7 \)
• When \( x = 0 \), \( f(0) = 2(0) - 5 = -5 \)
• When \( x = 1 \), \( f(1) = 2(1) - 5 = -3 \)
• When \( x = 2 \), \( f(2) = 2(2) - 5 = -1 \)

By doing these calculations and organizing them into a table, it's easy to spot the linear pattern showing the consistent increase in \( f(x) \) values as \( x \) increases. This method provides a clear visual of the function's behavior over a range of \( x \) values.

Graphical Representation

Graphing linear functions is a powerful way to visualize how they behave. The function \( f(x) = 2x - 5 \) can be graphically depicted by plotting points and connecting these points with a line. Each point corresponds to an \( x \) value from the previously created numerical table and its evaluated \( f(x) \) result.

Start by plotting the points calculated earlier:

• \((-2, -9)\)
• \((-1, -7)\)
• \((0, -5)\)
• \((1, -3)\)
• \((2, -1)\)

Once these points are plotted on a coordinate graph, a straight line passing through them illustrates the function. The slope of 2 indicates that for each step to the right (increase in \( x \)), the line rises by 2 units. The y-intercept, where the line crosses the y-axis, is at \( -5 \).

This line confirms the idea that \( f(x) \) consistently increases with \( x \), visualizing the function's linear nature. Graphs are excellent tools for understanding both trends and specific values at a glance.

Function Evaluation

Function evaluation involves substituting a specific value into the function to find the outcome or output. To verify the integrity of our representations, especially to check our calculations or the behavior of the function at specific points, evaluating a function is key. Let's examine \( f(x) = 2x - 5 \).

In this example, we're tasked with evaluating \( f(2) \). Start by substituting \( x \) with 2:

• Calculate: \( f(2) = 2(2) - 5 \)
• Solve: \( 4 - 5 = -1 \)

Therefore, \( f(2) = -1 \).

This result is key because it reaffirms our data from the numerical table and provides a single, precise inquiry into the function's mechanics. Function evaluation can be applied to any \( x \) value, allowing you to understand specific outputs and better grasp the entire function. The practice of evaluating helps build confidence in your analytical abilities, especially when cross-verifying with numerical and graphical representations.