Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window. Zoom out far enough to see the right-hand and left-hand behavior of each graph. Do the graphs of \(f\) and \(g\) have the same right-hand and Ieft- hand behavior? Explain why or why not. \(f(x)=3 x^{3}-9 x+1, \quad g(x)=3 x^{3}\)

Step-by-Step Solution

Answer

Yes, the graphs of \(f(x)\) and \(g(x)\) have the same right-hand and left-hand behavior because both increase without bound as \(x\) approaches positive infinity and decrease without bound as \(x\) approaches negative infinity. This is due to their leading term \(3x^3\).

Step 1: Graphing the functions

Use a graphing utility to plot the graphs of the functions \(f(x) = 3x^3 - 9x + 1\) and \(g(x) = 3x^3\). Be sure to zoom out enough to study the right-hand (as \(x\) approaches positive infinity) and left-hand (as \(x\) approaches negative infinity) behavior.

Step 2: Observing the right-hand and left-hand behaviors

Observe the right-hand and left-hand behavior of both graphs. You should find that both graphs increase without bound as \(x\) approaches positive infinity and decrease without bound as \(x\) approaches negative infinity. This shows that \(f\) and \(g\) have the same right-hand and left-hand behavior.

Step 3: Understanding the Observation

The reason for this behavior is that for both \(f(x)\) and \(g(x)\), their leading term is \(3x^3\), which determines the end behavior of the function. The other terms in \(f(x)\) influence the shape of the graph, but do not impact the right-hand and left-hand behavior.

Key Concepts

Right-Hand Behavior

The right-hand behavior of a polynomial function describes how the function behaves as the input variable, denoted by \( x \), approaches positive infinity. In simpler terms, it involves observing what happens to the value of the function when \( x \) gets very large. When graphing polynomial functions like \( f(x) = 3x^3 - 9x + 1 \) and \( g(x) = 3x^3 \), the right-hand behavior can be determined primarily by the leading term of the polynomial. In this case, both functions share the same leading term, \( 3x^3 \), which means they will exhibit similar behavior as \( x \) approaches positive infinity.

• For \( f(x) \), as \( x \to \, \infty \), \( f(x) \) increases without bound because of the positive leading coefficient \( 3 \) in the term \( 3x^3 \).
• Similarly, for \( g(x) \), \( g(x) = 3x^3 \) will also increase without bound as \( x \to \, \infty \).

Therefore, when examining the graphs of \( f \) and \( g \) using a graphing utility, you can observe that both functions "go upwards" as \( x \) becomes larger and larger.

Left-Hand Behavior

The left-hand behavior of a polynomial function deals with how the function behaves as \( x \) approaches negative infinity. It focuses on the pattern or direction the graph follows as \( x \) gets very small.For the functions \( f(x) = 3x^3 - 9x + 1 \) and \( g(x) = 3x^3 \), observing their left-hand behavior involves examining how their leading term, \( 3x^3 \), influences the graph as \( x \to -\infty \).

• For \( f(x) \), as \( x \to -\infty \), the function decreases without bound. The negative values of \( x \) make the \( 3x^3 \) term significantly negative due to the cubing process, causing the function to drop.
• Similarly, \( g(x) = 3x^3 \) exhibits the same left-hand behavior, decreasing without bound as \( x \to -\infty \).

This similarity in behavior happens because the term \( 3x^3 \) overrides any other terms when \( x \) is very large in magnitude. Thus, both graphs will "go downwards" as \( x \) becomes increasingly negative.

End Behavior of Functions

The end behavior of polynomial functions refers to the trend or direction the function takes as \( x \) moves towards positive or negative infinity. It helps in predicting how a function behaves in the extreme ranges of \( x \), a useful insight for sketching graphs and understanding their overall shape.In the case of our given functions, \( f(x) = 3x^3 - 9x + 1 \) and \( g(x) = 3x^3 \), the end behavior can be entirely attributed to their leading term, which is \( 3x^3 \).

• Right-hand end behavior: Both \( f(x) \) and \( g(x) \) increase indefinitely as \( x \to \, \infty \). This is derived from the positive leading coefficient of the \( x^3 \) term.
• Left-hand end behavior: Both functions decrease indefinitely as \( x \to -\infty \), which is again due to the influence of their main cubic term.

While the additional terms in \( f(x) \) influence its graph's nuances, they do not change the fundamental end behavior as modeled by \( 3x^3 \). Therefore, both functions share the same end behavior patterns, and this is crucial when you need to determine the overall direction of a function's graph.