Problem 26 - A Graphical Approach to College Algebra

Question

Graph each function in the standard viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. Over the interval specified, this function is __________. $$f(x)=x^{4} ;(0, \infty)$$

Step-by-Step Solution

Answer

Increasing

Step 1: Understanding the function

The function given is \( f(x) = x^4 \). This is a polynomial function with the term \( x^4 \) being the highest power of \( x \). The graph of this function is a curve that has a minimum point at \( x = 0 \) and extends upwards forever, forming a shape similar to the letter 'U'.

Step 2: Analyze the interval

The interval given is \((0, \infty)\). This means we are interested in the behavior of the function for \( x > 0 \). Since \( x^4 \) is a positive exponent and \( x > 0 \), the function outputs will be positive and increase as \( x \) increases.

Step 3: Graph the function

In a standard viewing window, typically from \( -10 \) to \( 10 \) on both axes, plot the graph of \( f(x) = x^4 \). For \( x > 0 \), observe that the function moves steeply upwards as \( x \) increases.

Step 4: Tracing the function

As you trace the graph from left to right starting at \( x = 0 \), you'll observe the function's values increase continuously, indicating that the function is getting larger as \( x \) increases.

Step 5: Conclusion about the interval

Since the function \( f(x) = x^4 \) consistently increases for every \( x > 0 \), we conclude that the function is increasing over the interval \((0, \infty)\).

Key Concepts

Graphing Functions

Graphing functions is an essential tool in understanding how functions behave visually. For polynomial functions like the one in our exercise, their graph usually forms smooth, continuous curves. For the given function, \( f(x) = x^4 \), its graph is shaped like a wide letter 'U'. This happens because the term \( x^4 \) is positive for all values of \( x \), which ensures the function's outputs are never negative. When you prepare to graph such functions using your calculator, make sure the viewing window settings can display the key features adequately. A typical setting is from -10 to 10 on both axes, which is usually good enough for observing general shapes and trends.
As you plot the graph of \( f(x) = x^4 \), notice how it swoops down to touch the x-axis at the origin \((x = 0)\) and then rises on both sides. However, our interval specification focuses on the part of the graph where \( x > 0 \). Thus, we only need to look at the right-hand section of this 'U', starting from the origin and moving upwards to the right.

Interval Analysis

When we perform interval analysis, we examine how a function behaves over specific portions of its domain. This helps in identifying patterns over those segments, such as increases or decreases.
In this exercise, we look at the interval \((0, \infty)\). Essentially, this means we're interested in the behavior of \( f(x) = x^4 \) for all positive values of \( x \). Interval analysis is crucial because it allows us to see how the function responds as the variable \( x \) moves through specified ranges. This polynomial function, having even degree, starts from minimal value at zero and climbs higher as we move in the positive direction. Given that polynomial functions like \( x^4 \) depend heavily on their leading term when \( x \) becomes large, it implies that the results grow large quite rapidly.

Increasing and Decreasing Functions

Polynomial functions can display sections where they continuously increase or decrease. Recognizing these trends is useful for interpreting the graph's behavior over certain intervals.
For \( f(x) = x^4 \), on the interval \((0, \infty)\), the function clearly is increasing. This is because, as \( x \) grows larger in value, the output \( f(x) \) also becomes larger, climbing ever upwards on the right side of the graph.To establish if a function is increasing, we observe if for any chosen \( x_1 < x_2 \), it holds that \( f(x_1) < f(x_2) \). For \( f(x) = x^4 \), for any \( x > 0 \), as \( x \) increases, \( x^4 \) naturally follows, which fits our earlier observations. Understanding whether a function is increasing or decreasing becomes helpful, especially in calculus and real-world applications, to determine growth trends.