Problem 113 - College Algebra

Question

For the following exercises, use the functions \(f(x)=-0.1 x+200\) and \(g(x)=20 x+0.1\). Where is \(f(x)\) greater than \(g(x) ?\) Where is \(g(x)\) greater than \(f(x)\) ?

Step-by-Step Solution

Answer

\(f(x)>g(x)\) for \(x < 9.95\); \(g(x)>f(x)\) for \(x > 9.95\).

Step 1: Identify Equations for Ine qualities

To find where one function is greater than the other, we need to set up the inequalities: - To find where \(f(x) > g(x)\), we need \(-0.1x + 200 > 20x + 0.1\). - To find where \(g(x) > f(x)\), we need \(20x + 0.1 > -0.1x + 200\).

Step 2: Solve Inequality for \(f(x) > g(x)\)

Rearrange the inequality \(-0.1x + 200 > 20x + 0.1\). Add \(0.1x\) and subtract \(0.1\) from both sides to get: \[200 - 0.1 > 20x + 0.1x\]This simplifies to: \[199.9 > 20.1x\]Solving for \(x\): \[x < \frac{199.9}{20.1} \approx 9.95\]

Step 3: Solve Inequality for \(g(x) > f(x)\)

Rearrange the inequality \(20x + 0.1 > -0.1x + 200\). Add \(0.1x\) and subtract \(0.1\) from both sides to get:\[0 > -20.1x + 199.9\]Solving for \(x\): \[x > \frac{199.9}{20.1} \approx 9.95\]

Step 4: Interpret the Solutions

The solution \(x < 9.95\) shows where \(f(x) > g(x)\). The solution \(x > 9.95\) shows where \(g(x) > f(x)\). At \(x = 9.95\), both functions are equal, meaning \(f(x) = g(x)\).

Key Concepts

Linear Functions

Linear functions are the building blocks of algebra and are used to describe a wide range of real-world phenomena. A linear function is a mathematical expression that creates a straight line when graphed on a coordinate plane. It's represented by the general formula:
\[f(x) = mx + b\]
where:

• \(m\) is the slope of the function, indicating its steepness and direction. A positive slope means the line rises, a negative slope means it falls.
• \(b\) is the y-intercept, the point where the line crosses the y-axis. It indicates the value of the function when \(x = 0\).

For example, in the function \(f(x) = -0.1x + 200\), the slope (\(-0.1\)) shows that the line decreases slightly as \(x\) increases. The y-intercept (200) means the line crosses the y-axis at 200, showing the starting value of the function when \(x\) equals zero. Similarly, for \(g(x) = 20x + 0.1\), a slope of 20 indicates a much steeper incline as \(x\) increases, starting near zero with the y-intercept at 0.1. These characteristics are crucial for understanding and comparing linear functions.

Inequality Solutions

Solving inequalities is a key skill in algebra. It involves finding the values of \(x\) that make an inequality true. Inequalities are similar to equations but instead of an equal sign, they use symbols like \(>\), \(<\), \(\geq\), or \(\leq\).
To solve inequalities, you perform similar steps as solving equations:

• Isolate \(x\) to one side of the inequality.
• Perform arithmetic operations, remembering to flip the inequality sign if you multiply or divide by a negative number.
• Simplify the inequality to find the solution for \(x\).

From the original exercise, we had two inequalities derived from the functions:

• To find \(x\) where \(f(x) > g(x)\), solve \(-0.1x + 200 > 20x + 0.1\). By isolating \(x\), it results in \(x < 9.95\).
• To find \(x\) where \(g(x) > f(x)\), solve \(20x + 0.1 > -0.1x + 200\). This simplifies to \(x > 9.95\).

These solutions tell us the range of \(x\) values where one function is greater than the other.

Function Comparison

Comparing two functions is essential to understand how they interact over different intervals. This involves analyzing where one function may be larger or smaller than another based on the values they produce for the same \(x\) inputs.
In the given exercise, we are tasked with comparing two linear functions, \(f(x)\) and \(g(x)\). The solutions highlight:

• Where \(f(x)\) is greater than \(g(x)\), which occurs when \(x < 9.95\). This indicates that function \(f\) outputs higher values than function \(g\) before reaching \(x = 9.95\).
• Where \(g(x)\) becomes greater than \(f(x)\), happening once \(x > 9.95\). Here, function \(g\) exceeds function \(f\) in terms of output values.
• At \(x = 9.95\), both functions produce the same output, meaning \(f(x) = g(x)\).

This comparison allows us to visualize and interpret the behavior of these functions over a range of \(x\), making it easier to apply in various contexts, such as predicting trends or making decisions based on function outputs.