Problem 25 - Algebra Form and Function

Question

In Table 4.8 , for which values of \(x\) is (a) \(f(x)>g(x) ?\) (b) \(\quad f(x)=g(x) ?\) (c) \(f(x)=0 ?\) (d) \(g(x)=0\) ? $$ \begin{array}{c|c|c|c|c|c|c|c|c} \hline x & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 4 & 1 & 0 & 1 & 4 & 9 & 16 & 25 \\ \hline g(x) & 1 / 4 & 1 / 2 & 1 & 2 & 4 & 8 & 16 & 32 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

(a) \(f(x) > g(x)\), (b) \(f(x) = g(x)\), (c) \(f(x) = 0\), and (d) \(g(x) = 0\). Answer: (a) \(f(x) > g(x)\) for \(x = -2, -1, 1, 2\). (b) \(f(x) = g(x)\) for \(x = 3, 4\). (c) \(f(x) = 0\) for \(x = 0\). (d) There are no values of \(x\) for which \(g(x) = 0\).

Step 1: (a) Find the values of \(x\) for which \(f(x) > g(x)\)

To answer this question, we will compare the values of \(f(x)\) and \(g(x)\) for each value of \(x\) in the given table. If \(f(x)\) is greater than \(g(x)\), we will note the value of \(x\). Comparing the values of the two functions, we find \(f(x) > g(x)\) for \(x=-2\), \(x=-1\), \(x=1\), and \(x=2\).

Step 2: (b) Find the values of \(x\) for which \(f(x) = g(x)\)

To answer this question, we will compare the values of \(f(x)\) and \(g(x)\) again. If \(f(x)\) is equal to \(g(x)\), we will note the value of \(x\). Comparing the values of the two functions, we find \(f(x) = g(x)\) for \(x=3\) and \(x=4\).

Step 3: (c) Find the values of \(x\) for which \(f(x) = 0\)

To answer this question, we will look for the values of \(x\) in the given table for which \(f(x) = 0\). Looking at the values of \(f(x)\), we find that \(f(x) = 0\) for \(x = 0\).

Step 4: (d) Find the values of \(x\) for which \(g(x) = 0\)

To answer this question, we will look for the values of \(x\) in the given table for which \(g(x) = 0\). However, looking at the values of \(g(x)\), we can conclude that there are no values of \(x\) in the table for which \(g(x) = 0\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics. It involves finding the output of a function for a particular input value. Imagine a function as a machine. You put in a value (the input), and the machine gives you back another value (the output). The output depends on the rule defined by the function.

Evaluating a function is straightforward. Suppose you have a function \( f(x) \). To find the value of \( f \) at, say, \( x = 2 \), you substitute \( 2 \) into the function in place of \( x \). The output is what you get after performing the operation described by the function.

• Understand the function's rule, like \( f(x) = x^2 \).
• Substitute the input value into the function, such as finding \( f(2) = 2^2 = 4 \).
• The result is the function's output at that input value.

Function evaluation is a foundational skill, crucial for analyzing how a function behaves at specific points.

Table Interpretation

Tables are a fantastic way to organize data effectively, making it easier to compare values and see relationships. In mathematics, tables often list values for different variables, showing how they change together.

In the context of function comparison, a table can help you evaluate and compare the outputs of two functions at given input values. In our example, we have a table listing values of \( x \), \( f(x) \), and \( g(x) \). By examining each row, you can determine different relationships between the functions: when one function is greater, equal, or less than the other.

• Read each row to see the corresponding values of \( f(x) \) and \( g(x) \) for the same \( x \).
• Compare the function values to each other.
• Note patterns or specific points of interest like intersections where \( f(x) = g(x) \).

Interpreting tables helps provide insight into function behaviors, simplifying what might otherwise be complex data sets.

Zeroes of a Function

Zeroes or roots of a function are the input values that produce an output of zero. Finding zeroes is critical because they often represent important points where a function crosses or touches the x-axis in a graph.

To find these zeroes in a given dataset or table, you need to look for entries where the function value is zero. For instance, if we are considering \( f(x) \), you check each \( x \) value in the table and see if it corresponds to a zero output.

• Examine the table for any rows where \( f(x) = 0 \).
• Write down the corresponding \( x \) values as the zeroes.
• If a function has multiple zeroes, note each one.

Finding the zeroes of a function is a key step in understanding its graphical and analytical properties, telling us about its intersections with the x-axis and the potential behavior of the graph.