Problem 64 - Intermediate Algebra

Question

Explain why the graph of \(f(x)=3^{x}\) gets closer and closer to the \(x\) -axis as the values of \(x\) decrease. Does the graph ever cross the \(x\) -axis? Explain why or why not.

Step-by-Step Solution

Answer

The graph of \( f(x) = 3^x \) gets closer to the \( x \)-axis as \( x \) becomes negative but never crosses it, because \( 3^x \) is never zero.

Step 1: Understanding Exponential Functions

Exponential functions are a key type of function in mathematics, which have the form \( f(x) = a^x \) where \( a \) is a constant. In the function \( f(x) = 3^x \), the base \( a \) is 3, which means the function grows or shrinks exponentially depending on \( x \).

Step 2: Behavior as \( x \) Decreases

As \( x \) becomes smaller (negative), the fraction \( 3^x = \frac{1}{3^{-x}} \) becomes smaller because the denominator \( 3^{-x} \) becomes very large. Therefore, \( f(x) = 3^x \) gets closer and closer to zero, which explains why the graph approaches the \( x \)-axis as \( x \) decreases.

Step 3: The Graph Never Crosses the \( x \)-axis

Since \( 3^x \) never actually becomes zero for any real \( x \), the graph approaches but never crosses the \( x \)-axis. This is because an exponential function like \( 3^x \) is never zero; it either grows indefinitely large as \( x \) increases or approaches zero (but never reaches it) as \( x \) decreases.

Step 4: Why the Graph Approaches Zero

Mathematically, for the expression \( 3^x \) to become zero, \( x \) would have to be negative infinity, which is not achievable in practical terms. No finite negative value of \( x \) will force \( 3^x \) to actually be zero; it only gets infinitesimally close.

Key Concepts

Graph of Exponential Function

The graph of an exponential function like \( f(x) = 3^x \) presents a unique and easily recognizable shape. An exponential function is defined by its constant base raised to a variable exponent. In this case, the base is 3. This function is characterized by a rapid increase or decrease depending on the value of the exponent \( x \).
The defining feature of exponential graphs is their distinct curvature. They tend to grow very quickly for positive \( x \) and shrink for negative \( x \). For \( f(x) = 3^x \), as \( x \) becomes more positive, the value of \( f(x) \) increases rapidly, resulting in a steep upward curve. Conversely, as \( x \) becomes negative, the function value decreases, creating a decay curve that approaches the x-axis without crossing it.

• Starts above the x-axis and never touches it.
• Steep growth as \( x \) increases.
• Approaches the x-axis as \( x \) decreases.

Behavior of Exponential Functions

Exponential functions exhibit distinct behaviors based on whether \( x \) is positive, negative, or zero. With \( f(x) = 3^x \), these behaviors can be seen quite clearly.
When \( x = 0 \), \( f(x) = 3^0 = 1 \). This tells us that the graph always passes through the point (0, 1).
For positive values of \( x \):

• As \( x \) increases, \( f(x) = 3^x \) increases very rapidly.
• The graph shows exponential growth, which can become extremely steep.

For negative values of \( x \):

• \( f(x) = 3^x \) translates into smaller fractional values like \( \frac{1}{3^{-x}} \).
• As \( x \) becomes more negative, the function value heads towards zero, showcasing exponential decay.
• The graph gets closer to zero without actually reaching it.

This uneven behavior is what makes exponential functions particularly interesting—they exhibit a smooth and continuous approach towards the axis without actually touching it.

X-axis Asymptote

The x-axis serves as a horizontal asymptote for the graph of \( f(x) = 3^x \). An asymptote in mathematics is a line that a graph approaches but never touches or crosses, except at infinity.
The reason the x-axis is an asymptote for \( 3^x \) lies in its behavior for negative \( x \). As we've established, as \( x \) decreases (becoming more negative), the value of \( f(x) = 3^x \) gets closer to zero.
Despite approaching the x-axis, \( 3^x \) never equals zero because exponents can only shrink \( 3^x \) so much. For any finite value of \( x \), \( 3^x \) remains positive, even if it's extremely small.

• The graph never crosses or touches the x-axis.
• This occurs because division by an increasingly large number results in a value approaching zero.
• Mathematically, the concept of \( 3^x = 0 \) is only a theoretical possibility at negative infinity.

Understanding the x-axis as an asymptote helps explain why exponential functions never touch or cross the x-axis, reinforcing the concept of limits within mathematical functions.