Problem 75 - A Graphical Approach to Precalculus with Limits

Question

\(f\) and \(g\) are linear functions. If the solution set of \(f(x) \geq g(x)\) is \([4, \infty),\) what is the solution set of each equation or inequality? (a) \(f(x)=g(x)\) (b) \(f(x)>g(x) \quad\) (c) \(f(x)

Step-by-Step Solution

Answer

(a) \(x = 4\), (b) \((4, \infty)\), (c) \((-\infty, 4)\).

Step 1: Understanding the initial condition

The inequality solution set \([4, \infty)\) tells us that for values of \(x\) starting at 4 and going to infinity, \(f(x) \geq g(x)\). This indicates at \(x = 4\), \(f(x) = g(x)\) because this is the boundary where \(f\) just starts to be greater than or equal to \(g\).

Step 2: Solving (a) for equality

Since \(f(4) = g(4)\), we have determined that the equation \(f(x) = g(x)\) has the solution \(x = 4\). This is the point where the transition happens from \(f(x) < g(x)\) to \(f(x) \geq g(x)\).

Step 3: Solving (b) for strict inequality

The inequality \(f(x) > g(x)\) holds true for all \(x > 4\). This means the solution set for \(f(x) > g(x)\) is \((4, \infty)\), as any value greater than 4 but not including 4 causes \(f(x)\) to be strictly greater than \(g(x)\).

Step 4: Solving (c) for less than relationship

Since \(f(x) < g(x)\) turns false at \(x = 4\), it must be true for values of \(x\) less than 4. Therefore, the solution set for \(f(x) < g(x)\) is \((-\infty, 4)\).

Key Concepts

Solution Set

The concept of a solution set might seem complex at first, but it's actually quite simple! When we solve an equation or inequality, we are finding all values of the variable that make the statement true. This collection of values is called a "solution set." For instance, in the exercise, the solution set for the inequality \(f(x) \geq g(x)\) is \([4, \infty)\). What does this mean?

• \([4, \infty)\) indicates that starting at \(x = 4\), and going to infinity, the statement \(f(x) \geq g(x)\) is true.
• The solution set includes 4 because the inequality is "greater than or equal to." This means right at \(x = 4\), \(f(x)\) is equal to \(g(x)\).
• All values greater than 4 make \(f(x)\) greater than \(g(x)\). Hence, the interval is open at infinity and closed at 4.

Understanding solution sets is crucial because it shows us all possible solutions, not just a single answer. This is particularly useful in understanding the behavior of functions over an interval.

Inequalities

Inequalities are mathematical expressions indicating that one quantity is larger or smaller than another. Unlike equations, which show equality, inequalities are about comparison and range.There are key symbols in inequalities:

• \(>\) means "greater than."
• \(<\) means "less than."
• \(\geq\) means "greater than or equal to."
• \(\leq\) means "less than or equal to."

In the exercise given, we have a range of inequalities involving \(f(x)\) and \(g(x)\):

• \(f(x) \geq g(x)\) indicates that for values starting at \(x = 4\), \(f(x)\) is at least equal to \(g(x)\).
• \(f(x) > g(x)\) indicates \(f(x)\) is strictly larger than \(g(x)\) for \(x > 4\).
• \(f(x) < g(x)\) is true for all values less than 4, meaning this inequality holds for negative infinity up to 4, but not including 4.

Solving inequalities helps us to understand the relationships between different functions and to visualize where one function overtakes another.

Linear Functions

A linear function is a type of function where the graph is a straight line. It can be represented by the equation \(y = mx + b\), where:

• \(m\) is the slope, indicating the steepness of the line.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

In the exercise, we are dealing with two linear functions, \(f(x)\) and \(g(x)\). Both functions can be graphed as lines:

• At \(x = 4\), \(f(x) = g(x)\), suggesting their lines intersect here.
• For \(x > 4\), \(f(x)\) is greater than \(g(x)\), meaning the slope or starting point of \(f(x)\) makes it rise above \(g(x)\).
• For \(x < 4\), \(g(x)\) is greater, revealing that \(g(x)\) overtakes \(f(x)\) before intersection.

Understanding linear functions and their graphs helps you to visually interpret the solution to inequalities and predict where certain conditions in equations or inequalities hold true.