Problem 30 - College Algebra

Question

Describe the end behavior of the graphs of the functions. $$f(x)=3\left(\frac{1}{2}\right)^{x}-2$$

Step-by-Step Solution

Answer

As \( x \to +\infty \), \( f(x) \to -2 \); as \( x \to -\infty \), \( f(x) \to \infty \).

Step 1: Identify the Function Type

The given function is \( f(x) = 3 \left(\frac{1}{2}\right)^{x} - 2 \). This is an exponential function, where the base of the exponent is \( \frac{1}{2} \), which is less than 1, indicating a decreasing function.

Step 2: Determine Horizontal Asymptote

For exponential functions of the form \( a \cdot b^{x} + c \), where \( 0 < b < 1 \), the horizontal asymptote is the constant \( c \). In this function, \( c = -2 \), so the horizontal asymptote is \( y = -2 \).

Step 3: Analyze End Behavior as \( x \to +\infty \)

As \( x \to +\infty \), the term \( \left(\frac{1}{2}\right)^{x} \to 0 \), so \( f(x) \to 3 \,\cdot\, 0 - 2 = -2 \). The graph approaches the horizontal asymptote \( y = -2 \) from above.

Step 4: Analyze End Behavior as \( x \to -\infty \)

As \( x \to -\infty \), the term \( \left(\frac{1}{2}\right)^{x} \to \infty \) because the exponent effectively flips the fraction. Hence, \( f(x) \to 3 \,\cdot\, (large \; positive \; value) - 2 \), meaning \( f(x) \to \infty \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where variables appear as exponents. In general terms, they are written as \( f(x) = a \cdot b^{x} + c \), where the base \( b \) is a positive real number. The behavior and the characteristics of the function largely depend on the value of the base.

• If \( 0 < b < 1 \), as it is in our function \( f(x) = 3 \left(\frac{1}{2}\right)^{x} - 2 \), the function is said to be a decreasing exponential function.
• Here, the base \( \frac{1}{2} \) indicates that the function value decreases as \( x \) increases.
• Conversely, if \( b > 1 \), it would represent an increasing exponential function.

Exponential functions are widely used to model processes that change rapidly, such as radioactive decay or population growth, depending on whether they are decreasing or increasing.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a function approaches as the input \( x \) becomes very large or very small. For exponential functions of the form \( f(x) = a \cdot b^{x} + c \), the horizontal asymptote is the constant \( c \).

• In our function \( f(x) = 3 \left(\frac{1}{2}\right)^{x} - 2 \), the horizontal asymptote is \( y = -2 \).
• As \( x \) tends towards \( +\infty \), the function value \( f(x) \) approaches this horizontal asymptote, \( y = -2 \), but never actually reaches it.

The horizontal asymptote can inform you about the long-term behavior of the function, which is crucial in understanding how a function behaves as \( x \) becomes very large or very small.

Decreasing Function Behavior

A function is considered decreasing when its value diminishes as the input \( x \) increases. For exponential functions, the base impacts whether the function will increase or decrease over its domain. In \( f(x) = 3 \left(\frac{1}{2}\right)^{x} - 2 \), the base \( \frac{1}{2} \) is less than 1, indicating a decreasing function.

• As \( x \to +\infty \), the term \( \left(\frac{1}{2}\right)^{x} \) approaches zero, making the whole function shrink toward the horizontal asymptote, \( y = -2 \).
• As \( x \to -\infty \), the function value grows significantly because the term \( \left(\frac{1}{2}\right)^{x} \) approaches exceedingly large numbers.

This shifting behavior between negative infinity and positive values helps in predicting how the exponential function behaves in different ranges and ensures clarity in its end behavior.