Problem 14 - College Algebra

Question

For the following exercises, determine whether each function is increasing or decreasing. $$ f(x)=4 x+3 $$

Step-by-Step Solution

Answer

The function is increasing.

Step 1: Recognize the Function Type

The given function is a linear function of the form \( f(x) = mx + b \), where \( m = 4 \) and \( b = 3 \). Linear functions are generally straightforward to analyze for increasing or decreasing behavior based on the slope \( m \).

Step 2: Analyze the Slope

The slope \( m \) of the linear function is 4. In linear functions, if \( m > 0 \), the function is increasing; if \( m < 0 \), the function is decreasing. Since \( m = 4 \), which is greater than 0, this function is increasing.

Step 3: Conclusion

Based on the positive slope \( m = 4 \), we can conclude that the function \( f(x) = 4x + 3 \) is an increasing function.

Key Concepts

Understanding Increasing Functions

An increasing function is one where the output (or function value) rises as the input (or variable) increases. Imagine you are climbing a hill; if every step you take moves you higher, you are on an increasing path.

In mathematical terms, for any two points \(x_1\) and \(x_2\) on the function, if \(x_1 < x_2\), then \(f(x_1) < f(x_2)\). In simpler words, as we move from left to right on the graph, the line goes up.

Why does this happen? Because of the slope of the function. If the slope is positive, then the function is increasing. In the context of linear functions, they can either be increasing, decreasing, or constant. An increasing linear function will always have a rising line from left to right across the graph.

Slope Analysis in Linear Functions

The slope of a linear function is a crucial piece of information. It's represented by \(m\) in the equation \(f(x) = mx + b\). Understanding the slope tells us how the function behaves across the x-axis.

Here are a few key points about slope analysis:

• If \(m > 0\), the function increases as you move from left to right.
• If \(m < 0\), the function decreases as you move from left to right.
• If \(m = 0\), the function remains constant, with no increase or decrease.

The steeper the slope, the more rapidly the function increases or decreases. In the exercise we had, the slope \(m = 4\), which means every step to the right increases the output by 4 units. This confirms the function is increasing.

The Simple Role of Algebra in Understanding Functions

Algebra is like the toolkit for understanding and solving function-related problems. It provides a structure to interpret mathematical relationships. With algebra, you can quickly decide if a function is increasing or decreasing by just looking at its formula.

In our exercise, we used the linear function formula, \(f(x) = mx + b\). By identifying the values of \(m\) and \(b\), algebra lets us analyze the slope without graphing. The beauty of algebra is its power to simplify complex ideas into understandable concepts, making it easier to understand and apply across math and science.

So, with these algebraic tools, deciding if functions are increasing becomes straightforward. You're looking at how algebra shapes the lines and how these lines tell a story about the relationship between variables.