Problem 3 - A Graphical Approach to College Algebra

Question

Provide a short answer to each question. Do not use a calculator. What is the largest open interval over which \(f(x)=\frac{1}{x}\) increases? decreases? is constant?

Step-by-Step Solution

Answer

Increases: None; Decreases: \((-\infty, 0) \cup (0, \infty)\); Constant: None.

Step 1: Determine Interval of Increase

The function \( f(x) = \frac{1}{x} \) increases when its derivative \( \frac{d}{dx} \left( \frac{1}{x} \right) = -\frac{1}{x^2} \) is positive. Since \( -\frac{1}{x^2} \) is always negative for all \( x eq 0 \), \( f(x) \) does not increase anywhere.

Step 2: Determine Interval of Decrease

The function \( f(x) = \frac{1}{x} \) decreases on intervals where its derivative \( -\frac{1}{x^2} \) is negative. This occurs for all \( x eq 0 \). Thus, \( f(x) \) decreases on the intervals \( (-\infty, 0) \) and \( (0, \infty) \).

Step 3: Determine Interval Where the Function is Constant

A function is constant on an interval if its derivative is zero over that interval. Since \( -\frac{1}{x^2} \) is never zero for any \( x eq 0 \), \( f(x) = \frac{1}{x} \) is not constant on any open interval.

Key Concepts

Interval of Increase

When we talk about the interval of increase for a function, we mean the intervals where the function's values are getting larger as the input values get larger. To find these intervals, we use the derivative of the function. The derivative tells us the rate at which the function's value is changing.

For the function \( f(x) = \frac{1}{x} \), the derivative is \( -\frac{1}{x^2} \). This expression is always negative for all \( x eq 0 \). This means that the function is always decreasing, not increasing, since negative derivatives indicate a downward slope. Thus, an interval of increase does not exist for this function.

Remember, an interval of increase can only be found where the derivative is positive, leading to a "climbing" pattern in the graph. For \( f(x) = \frac{1}{x} \), since this never happens, we must conclude there are no intervals where the function increases.

Interval of Decrease

The interval of decrease refers to the segments of a function where its value diminishes as the input increases. We determine this using the derivative: negative values indicate a decrease.

For \( f(x) = \frac{1}{x} \), its derivative is \( -\frac{1}{x^2} \), which remains negative for any \( x eq 0 \). Therefore, \( f(x) = \frac{1}{x} \) decreases across its entire domain, except at \( x = 0 \), where the function is undefined.

Thus, we identify that the intervals of decrease for \( f(x) \) are both \( (-\infty, 0) \) and \( (0, \infty) \). This means wherever you look on the graph, except right at \( x = 0 \), the function's value is dropping as you move along either interval.

Constant Function

A constant function maintains the same output regardless of input changes. This occurs when the derivative is zero, meaning no change across any interval.

For \( f(x) = \frac{1}{x} \), the derivative, \( -\frac{1}{x^2} \), is never zero except where \( x = 0 \), where the function itself is not defined. Since the derivative does not equal zero for any value of \( x \) in the domain of the function, there are no intervals where this function is constant.

To illustrate, imagine a flat, level surface visually represented on a graph, which is what a constant function would look like. However, \( f(x) = \frac{1}{x} \) always has some slope, so it never flattens out completely over any interval.