Problem 56 - Algebra and Trigonometry

Question

Functions That Are Always Increasing or Decreasing Sketch rough graphs of functions that are defined for all real numbers and that exhibit the indicated behavior (or explain why the behavior is impossible). (a) \(f\) is always increasing, and \(f(x)>0\) for all \(x\) (b) \(f\) is always decreasing, and \(f(x)>0\) for all \(x\) (d) \(f\) is always decreasing, and \(f(x)<0\) for all \(x\)

Step-by-Step Solution

Answer

(a) Use \( f(x) = e^x \). (b) Use \( f(x) = \frac{1}{x^2+1} \). (d) Use \( f(x) = -e^x \).

Step 1: Sketching Increasing Function That is Always Positive

To create a graph of a function that is always increasing and positive for all real numbers, consider the function \[ f(x) = e^x \] or \[ f(x) = x^2 + 1 \].Both functions fulfill two criteria:- They are always positive since \( e^x > 0 \) for all \( x \) and \( x^2 + 1 > 0 \) for all \( x \).- Both are always increasing. The exponential function \( e^x \) increases as \( x \) increases, and for \( x^2 + 1 \), if we consider \( x > 0 \), \( x^2 + 1 \) is increasing.

Step 2: Sketching Decreasing Function That is Always Positive

For a function that is always decreasing and positive, consider the function \[ f(x) = \frac{1}{x^2+1} \].This function is always positive because the denominator \( x^2+1 \) is always positive, and the function is always decreasing because as \( x \) increases, \( x^2+1 \) increases, causing \( \frac{1}{x^2+1} \) to decrease.

Step 3: Sketching Decreasing Function That is Always Negative

For a function that is always decreasing and negative, consider the function \[ f(x) = -e^x \].This function is always negative since \( e^x \) is always positive, making \( -e^x \) always negative. It is decreasing because \( e^x \) is increasing, but when negated, it results in a function that decreases as \( x \) increases.

Key Concepts

Increasing Functions

Understanding increasing functions is crucial to understanding how functions behave. An increasing function continuously rises as you move along the x-axis. More formally, a function \( f(x) \) is said to be increasing if for every two points, \( x_1 \) and \( x_2 \), where \( x_1 < x_2 \), it holds that \( f(x_1) < f(x_2) \). In simple terms, the y-values get bigger as you move from left to right along the x-axis.
A classic example of an increasing function is the exponential function \( f(x) = e^x \). Here:

• As \( x \) increases, \( e^x \) increases rapidly.
• This function is always positive because the exponential function never touches the x-axis.
• Another example is \( f(x) = x^2 + 1 \), but it's important only when \( x \) is evaluated for \( x > 0 \).

Both these functions are great examples of an ever-ascending curve, enabling you to easily visualize increasing behaviors.

Decreasing Functions

Decreasing functions are the opposite of increasing functions. For a function \( f(x) \) to be decreasing, it means that for any two possible points \( x_1 \) and \( x_2 \), such that \( x_1 < x_2 \), the condition \( f(x_1) > f(x_2) \) must be satisfied. In other words, the y-values decline as you progress from left to right along the x-axis.
Consider the function \( f(x) = \frac{1}{x^2 + 1} \). This is a perfect depiction of a decreasing function:

• As \( x \) grows, \( x^2 + 1 \) increases, which in turn, decreases the value of \( \frac{1}{x^2 + 1} \).
• This maintains positivity, and the line slides down as it moves right.
• Another decreasing function is \( f(x) = -e^x \), which is not only decreasing but also always negative.

These functions illustrate how decreasing functions slide downwards, reflecting their downward trajectory on a graph.

Exponential Function Behavior

Exponential functions like \( e^x \) are special due to their rapid growth properties. These functions have a form of \( f(x) = a^x \), where \( a > 0 \), and they serve to model phenomena that grow at an accelerating rate, much like populations or compound interest.
Some characteristics to note:

• Their rate of increase gets faster; they virtually "shoot up" as \( x \) increases.
• In their simplest form \( e^x \), they are always positive—never touching the x-axis.
• These functions are inherently increasing, maintaining positivity with huge practical applications in science and mathematics.

Thus, the exponential function behavior is vital for understanding not just mathematical concepts, but real-world applications that exhibit exponential growth.

Positive and Negative Functions

The concept of positive and negative functions hinges on whether the outcome of the function, \( f(x) \), is entirely above or below the x-axis.
A function is positive if all its values are greater than zero for all \( x \). Take \( f(x) = e^x \), which is always positive. Why?

• Regardless of \( x \), \( e^x > 0 \).

Conversely, consider \( f(x) = -e^x \). This function is always negative, mirroring the positive behavior of \( e^x \) below the x-axis:

• It remains \( < 0 \) for any \( x \), producing only negative values.

Understanding which parts of a function are positive or negative helps in graphing and predicting the behavior of physical systems that might oscillate between high and low output values.