Problem 13 - Precalculus : Building Concepts and Connections

Question

Use the following table of test values of the quadratic functions \(f\) and \(g\) defined on \((-\infty, \infty)\). $$\begin{array}{ccc} t & f(t) & g(t) \\ -1 & 0 & 2 \\ -0.5 & 0.5 & 1.25 \\ 0 & 1 & 1 \\ 0.5 & 1.5 & 1.25 \\ 1 & 2 & 2 \\ 1.5 & 2.5 & 3.25 \\ 2 & 3 & 5 \\ 2.5 & 3.5 & 7.25 \\ 3 & 4 & 10 \end{array}$$ Find the region(s) where \(f(t) \geq g(t)\).

Step-by-Step Solution

Answer

The functions \(f(t)\) and \(g(t)\) are such that \(f(t) \geq g(t)\) when \(t\) is in the interval [-1, 0.5] or equal to 1.

Step 1: Understand the table

First, look at the table and compare the \(f(t)\) and \(g(t)\) values for each \(t\) value. Write down in which 't' intervals \(f(t) \geq g(t)\).

Step 2: Identify where \(f(t) \geq g(t)\)

Based on the table, \(f(t) \geq g(t)\) when \(t\) is between -1 and 0.5, and when \(t\) is equal to 1.

Step 3: Write the final answer

Write down the answer as the 't' intervals in which \(f(t) \geq g(t)\) happens. Make sure to include both end points and the in-between interval.

Key Concepts

Function Inequality

Function inequalities involve comparing two functions to determine where one is greater than or equal to the other. In essence, for two functions \(f(t)\) and \(g(t)\), we want to find all values of \(t\) where the inequality \(f(t) \geq g(t)\) holds true.

When dealing with function inequalities:

• Look at specific points where their values are given.
• Compare values to determine which function is larger at each point.
• Identify intervals where the inequality is satisfied.

This process seems simple but is quite crucial in understanding the behavior of functions across different domains. By comparing function values, you unravel the regions where one function outperforms or equals the other.

Interval Notation

Interval notation is a mathematical notation used for representing a range of numbers. It's essential for expressing where a function inequality holds true.

For the solution:

• Use brackets \([\, ]\) for closed intervals, where the endpoint is included.
• Use parentheses \((\, )\) for open intervals, where the endpoint is not included.

For example, when we say \([-1, 0.5)\), it means \(t\) ranges from \(-1\) to \(0.5\) including \(-1\) but not 0.5. It's crucial to be accurate with intervals to accurately reflect the conditions where inequalities hold.

This notation makes it easy to identify exactly where in the domain the inequality is satisfied.

Quadratic Inequality

Quadratic inequalities involve expressions where a quadratic function is set to be greater or equal to, or less than another function. A quadratic function is in the form \(ax^2 + bx + c\).

To solve quadratic inequalities like \(f(t) \geq g(t)\):

• Calculate and compare values of the quadratic functions at given points.
• Look at test values from a table or graph.
• Identify those intervals where the inequality holds and express them using interval notation.

Understanding quadratic inequalities helps in analyzing solutions within certain ranges of values. It's foundational for grasping how quadratic functions behave in relation to each other. By learning these, you gain insight into the broader scope of quadratic functions and their inequalities.