Problem 90 - College Algebra with Modeling and Visualization

Question

If the graph of \(f\) is decreasing on \([0, \infty),\) then what can be said about the graph of \(y=f(-x)+17\) the graph of \(y=-f(x)-1 ?\)

Step-by-Step Solution

Answer

Both functions are increasing on \([0, \infty)\).

Step 1: Understand the Nature of the Function

The function \( f \) is given to be decreasing on the interval \([0, \infty)\). A function is decreasing if, for any two points \( x_1 < x_2 \), \( f(x_1) > f(x_2) \). This means that as \( x \) increases, \( f(x) \) decreases.

Step 2: Analyze the Function \( y = f(-x) + 17 \)

Since \( f \) is decreasing on \([0, \infty)\), it is also decreasing on \((-rac{x}{-1}, rac{x}{-1})\), which translates to \((-\infty, 0] \) when we consider \( f(-x) \). Thus, \( f(-x) \) is increasing on \([0, \infty)\) since \(-x \) makes inputs negative and effectively flips the direction of the interval for \( x \). Adding 17 does not affect the increasing nature; thus, \( y = f(-x) + 17 \) is increasing on \([0, \infty)\).

Step 3: Analyze the Function \( y = -f(x) - 1 \)

Since \( f(x) \) is decreasing on \([0, \infty)\), \( -f(x) \) will reverse the inequality, making \(-f(x)\) increasing on \([0, \infty)\). Subtracting 1 is a vertical shift downwards and does not affect the increasing nature; thus, \( y = -f(x) - 1 \) is increasing on \([0, \infty)\).

Key Concepts

Decreasing Function

A decreasing function is one where, as you move from left to right along the x-axis, the y-values get smaller. If a function is decreasing on an interval \([a, b]\), it means that for any two points \((x_1 < x_2)\) within this interval, \(f(x_1) > f(x_2)\). This can seem intuitive when comparing it to rolling a ball downhill—it never goes up, only stays flat or moves downward.
What happens in our specific problem? We have the function \(f\) that is said to be decreasing on \([0, \infty)\). This tells us that as \(x\) grows from 0 onwards, \(f(x)\) continuously decreases. Understanding this behavior is the foundation for analyzing how transformations affect the function.

Increasing Function

An increasing function is the opposite of a decreasing function. Here, as \(x\) increases, \(f(x)\) also increases. For this behavior to occur, for any two points \((x_1 < x_2)\) within the interval, \(f(x_1) < f(x_2)\). This can be visualized like climbing a hill—it only stays flat or goes upwards.
In the exercise, after flipping the input in \(f(-x)+17\) or multiplying the function by a negative in \(-f(x)-1\), both transformations result in increasing functions on the interval \([0, \infty)\). The original decreasing behavior is reversed by these operations. It’s like looking at a graph in a mirror or inverting a roller coaster.

Function Transformation

Transformation of functions involves changing their position or shape on the graph by performing operations on them. Common transformations include reflections, translations, and stretches.

• Reflection: In \(f(-x)+17\), the negative sign inverts the x-values, causing a horizontal reflection across the y-axis. This changes a decreasing function into an increasing one.
• Vertical Translation: Adding a constant like 17 results in moving the whole function upward by 17 units, which modifies the graph's position without affecting its increasing or decreasing nature.
• Multiplying by a Negative: When the function becomes \(-f(x)-1\), multiplying by -1 reflects it across the x-axis. This changes decreasing to increasing, and subtracting 1 shifts it downward.

These transformations affect both the appearance and behavior of the function, and understanding them is crucial for analyzing any graph effectively.