Problem 27 - Algebra Form and Function

Question

Figure 7.6 shows that the graph of \(y=x^{4}\) is above the graph of \(y=x^{2}\) when \(x\) is greater than \(1,\) and it is below it when \(x\) is between 0 and \(1 .\) Express these facts algebraically using inequalities.

Step-by-Step Solution

Answer

Question: Determine the inequalities that represent the relationship between the functions \(y = x^4\) and \(y = x^2\) for the domains \(x > 1\) and \(0 < x < 1\). Answer: \(x^4 > x^2\) when \(x > 1\), and \(x^4 < x^2\) when \(0 < x < 1\).

Step 1: Analyse the given graphs

Comparing the graphs of \(y = x^4\) and \(y = x^2\) as stated in the exercise. Notice that the given graph \(y = x^4\) is entirely above the graph of \(y = x^2\) when \(x>1\). Then, the graph \(y = x^4\) is below \(y = x^2\) when \(0 < x < 1\).

Step 2: Determine inequality when \(x > 1\)

As we found in the first step, the graph of \(y = x^4\) is above \(y = x^2\) when \(x > 1\), so we want to find the inequality that represents this relationship. To do this, we compare both expressions and find out which one is greater: \(x^4 > x^2\) when \(x > 1\).

Step 3: Determine inequality when \(0 < x < 1\)

In the first step, we also found that the graph of \(y = x^4\) is below \(y = x^2\) when \(0 < x < 1\). So, let's compare both expressions and find the inequality that represents this case: \(x^4 < x^2\) when \(0 < x < 1\). So, we've successfully broken down the exercise into smaller steps to solve the problem. The two inequalities represent the given information algebraically: 1. \(x^4 > x^2\) when \(x > 1\). 2. \(x^4 < x^2\) when \(0 < x < 1\).

Key Concepts

Graphing Functions

Graphing functions is an essential part of understanding the behavior of mathematical expressions visually. It helps to compare different functions by observing where their graphs lie relative to each other on a coordinate grid. For the exercise given, we are observing the graphs of two functions: \(y = x^4\) and \(y = x^2\). By plotting these functions, you can easily see at which intervals one graph is above or below the other.

To graph a function:

• Choose a range for \(x\) values.
• Calculate the corresponding \(y\) values using the function's formula.
• Plot the points \((x, y)\) on a coordinate plane.
• Connect the points to see the shape of the graph.

The visual comparison of graph positions allows us to deduce inequalities between functions within different ranges of \(x\). This graphical insight is crucial, especially when dealing with polynomial functions.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number exponents. For example, both \(y = x^4\) and \(y = x^2\) belong to this family. These functions usually exhibit smooth and continuous curves when graphed. They can have characteristics such as being symmetric around the y-axis or having roots, maxima, and minima.

Key features of polynomial functions include:

• Degree: The highest exponent of the variable e.g., 4 for \(x^4\).
• Coefficients: The numeric factors multiplied by the variable e.g., 1 in \(x^4\).
• End behavior: How the graph behaves as \(x\) approaches positive or negative infinity.

Understanding these features can help predict and analyze the behavior of the graph within different intervals. For the exercise, recognizing that \(x^4\) grows faster than \(x^2\) demonstrates why \(y = x^4\) is above \(y = x^2\) when \(x > 1\). Conversely, \(x^4\) grows slower than \(x^2\) between \(0 < x < 1\), causing it to be below.

Comparing Functions

Comparing functions involves analyzing their graphs or algebraic expressions to establish which is greater, lesser, or equal in certain intervals. This is done by determining the inequalities governing their relationship across different ranges of \(x\).

When comparing \(y = x^4\) and \(y = x^2\):

• For \(x > 1\), since \(x^4 > x^2\), \(y = x^4\) is higher.
• For \(0 < x < 1\), since \(x^4 < x^2\), \(y = x^4\) is lower.

This analysis leads to inequalities that define their relationship:

• \(x^4 > x^2\) when \(x > 1\)
• \(x^4 < x^2\) when \(0 < x < 1\)

By understanding these differences, you can solve problems involving function comparisons and predict outcomes based on changes in \(x\). This process of comparison reveals not just the ordering, but also critical points of intersection and turning points in the graphs of polynomial functions.