91Ó°ÊÓ

In mathematics, understanding the domain and range of a function is crucial.The domain of a function is the complete set of possible input values (x-values) for which the function is defined. For exponential functions like \( f(x) = 10^{-x} \), the domain is all real numbers. This means you can plug in any real number for \( x \), and the function will provide a result.

• Domain: The domain of \( f(x) = 10^{-x} \) is \( (-\infty, \infty) \).

The range, on the other hand, is all the possible output values (y-values) that the function can produce. For exponential decay functions like this one, the function values are positive real numbers. This means the function never actually reaches zero or becomes negative.

• Range: For \( f(x) = 10^{-x} \), the range is \( (0, \infty) \).

Understanding these concepts helps in sketching the graph and predicting the behavior of the function across different values of \( x \).

Exponential functions often feature a horizontal asymptote. An asymptote is a line that the graph of a function approaches but never touches.For \( f(x) = 10^{-x} \), as \( x \to \infty \), the value of \( f(x) \) becomes increasingly small, approaching zero but never actually reaching it. Hence, the horizontal asymptote is the x-axis.

• The equation of the horizontal asymptote is: \( y = 0 \).

This concept is crucial because it helps in understanding the limits of the function's output as the input grows larger or smaller. The presence of a horizontal asymptote indicates that no matter how large \( x \) grows, the function's value will never equal zero, only approaching it infinitesimally closely.

Exponential decay functions describe situations where quantities decrease rapidly at a rate proportional to their current value.In the function \( f(x) = 10^{-x} \), the negative exponent indicates exponential decay. As \( x \) increases, \( 10^{-x} \) becomes very small, which shows the function is decreasing.

• Behavior: The function decreases as the value of \( x \) increases.

Key markers in the function's graph include:- At \( x = 0 \), \( f(x) = 10^0 = 1 \).- At \( x = 1 \), \( f(x) = 10^{-1} = 0.1 \).- As \( x \to \infty \), \( 10^{-x} \to 0 \), but never actually reaches zero.Understanding exponential decay is important in various fields, from physics to finance, where it explains phenomena like radioactive decay or depreciation of assets. Recognizing the characteristic downward curve is essential for interpreting graphs and data linked with this concept.