Problem 13 - The Calculus with Analytic Geometry

Question

\(f(x)= \begin{cases}-x^{4} & \text { if } x<0 \\ x^{4} & \text { if } x \geq 0\end{cases}\)

Step-by-Step Solution

Answer

The function is \(f(x) = -x^4 \) for \(x < 0\) and \(f(x) = x^4 \) for \(x \geq 0\).

Step 1: Identify the Function

The function is given in two pieces: - For values of x less than 0, the function is defined as \(f(x)=-x^4\). - For values of x greater than or equal to 0, the function is defined as \(f(x)=x^4\).

Step 2: Determine the Function Behavior for x < 0

When x is less than 0, the function follows the formula \(f(x)=-x^4\). This implies that for any negative x, the output will be the negative of x raised to the fourth power.

Step 3: Determine the Function Behavior for x ≥ 0

When x is greater than or equal to 0, the function follows the formula \(f(x)=x^4\). This implies that for any non-negative x, the output will be x raised to the fourth power.

Step 4: Visualize the Function

For better understanding, it might help to sketch the graph of the function. The left part (x < 0) will be a downward curve \(f(x) = -x^4 \), and the right part (x ≥ 0) will be an upward curve \(f(x) = x^4 \). Both curves meet at \(x = 0\) where \(f(0) = 0 \).

Key Concepts

Function Behavior

To understand the behavior of a piecewise function, it's essential to look at each piece separately. In our example, we have:

• For values of x less than 0, the function is defined as \( f(x) = -x^4 \). Since \( x^4 \) is always positive for any real number, multiplying by -1 makes it negative. So, for negative x, the function dips below the x-axis.
• For values of x greater than or equal to 0, the function is defined as \( f(x) = x^4 \), resulting in all non-negative outputs. This means the function rises above the x-axis.

By studying each segment, we can predict the overall shape of the function effectively.

Graph Visualization

A visual representation of a function can make it much easier to grasp. For our given function, plotting the points and drawing the curves will help us see the behavior clearly.Starting with \( f(x) = -x^4 \) for \( x < 0 \), we get a downward curve because raising a negative number to the fourth power gives a positive result, but the negative sign in front flips it downward. This curve starts at the origin and goes down.On the right side, where \( x \geq 0 \), the function is \( f(x) = x^4 \). Here, the curve is upward, starting from the origin and rising up. Remember:

• Both parts of the function are symmetric around the y-axis because of the even power (4).
• The point where they meet is at \(x = 0\) and the value is 0 as \(0^4 = 0\).

This understanding helps in sketching accurate graphs and in predicting how the function behaves at different intervals.

Continuity of Functions

A function's continuity can be examined by checking if there are any breaks, jumps, or holes in its graph. For the given piecewise function:

• If \( x < 0 \), the function follows \( f(x) = -x^4 \).
• If \( x \geq 0 \), the function follows \( f(x) = x^4 \).

Because both pieces meet seamlessly at \( x = 0 \) with \( f(0) = 0 \), the function is continuous at that point. No breaks, jumps, or holes are present since the value from both sides approaches the same point (0,0). Hence, the function is continuous everywhere on its domain.
Understanding continuity is crucial for analyzing the behavior and properties of functions in calculus and analysis.