Problem 56 - 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 \(f(2).\) $$ f(x)=8 $$

Step-by-Step Solution

Answer

The value of \( f(2) \) is 8.

Step 1: Introduction to the Function

The function given is a constant function, which means it assigns the same output value for every input value. Here, the function is given by \( f(x) = 8 \), indicating that for any value of \( x \), \( f(x) \) equals 8.

Step 2: Verbal Representation

The verbal representation of the function \( f(x) = 8 \) is that it is a constant function where for every value of \( x \), \( f(x) \) will always be 8. This means there is no change in the output regardless of the input.

Step 3: Graphical Representation

Graphically, since the function is \( f(x) = 8 \), it will be represented as a horizontal line on the graph at \( y = 8 \). This line runs parallel to the x-axis because the value of the function does not change.

Step 4: Numerical Representation Table

To create a numerical representation, we will evaluate the function for specified values of \( x \): -2, -1, 0, 1, and 2. Since \( f(x) = 8 \) for any x-value, the table will look like this:| x | f(x) ||-----|------||-2 | 8 ||-1 | 8 || 0 | 8 || 1 | 8 || 2 | 8 |

Step 5: Evaluation of f(2)

To evaluate \( f(2) \), substitute \( x=2 \) into the function. Since \( f(x) = 8 \) for any \( x \), \( f(2) \) simply equals 8.

Key Concepts

Constant Function

A constant function is a special type of function in algebra that outputs the same value, no matter what input you plug into it. This is what happens with our function, where for every input, the output is guaranteed to be 8. The general form of a constant function is expressed as \( f(x) = c \), where \( c \) is a constant. In our case, \( c \) is 8. Understanding constant functions is crucial because they signify stability and predictability. If you had to rely on a specific outcome every time, like a fixed interest rate or a subscription fee, you would be encountering a constant function. They help in simplifying computations since there's no change in the output relative to changing the input.

Graphical Representation

In mathematics, visualizing a function can often make it much easier to understand. For a constant function like \( f(x) = 8 \), its graphical representation is straightforward. Imagine plotting a horizontal line across your graph at the y-value of 8. This line runs parallel to the x-axis and does not cross it, emphasizing that no matter the input value of \( x \), the output \( f(x) \) remains 8. This graphical simplicity also tells you that the function does not have a slope or angle—it’s purely horizontal. Such a graph helps in interpreting the behavior of the function at a glance. For those looking for stability or constancy, the horizontal line is an indicator that nothing changes as inputs vary across the x-axis.

Numerical Representation

Numerical representations condense a function's behavior into a table format, providing clear insight into its consistency. For the function \( f(x) = 8 \), creating a table with specific values of \( x \) gives a very clear picture. The values chosen here are \( x = -2, -1, 0, 1, 2 \). For each of these inputs, the function consistently outputs 8:

• \( x = -2 \), \( f(x) = 8 \)
• \( x = -1 \), \( f(x) = 8 \)
• \( x = 0 \), \( f(x) = 8 \)
• \( x = 1 \), \( f(x) = 8 \)
• \( x = 2 \), \( f(x) = 8 \)

This table emphasizes that no matter the input, the outcome is constant. Such a representation is useful for problem-solving or when summarizing function behavior, as it provides a quick reference that the function is always 8, reflecting the constancy and predictability of this function type.