Problem 28 - A Graphical Approach to College Algebra

Question

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=10^{-x}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), range: \((0, \infty)\), asymptote: \(y=0\). Function is decreasing.

Step 1: Understand the Function

The function given is an exponential function in the form of \( f(x) = 10^{-x} \). The base of the exponential is 10, but since it's raised to the power of \(-x\), the function exhibits exponential decay.

Step 2: Identify the Domain and Range

The domain of any exponential function is all real numbers, so the domain of \( f(x) = 10^{-x} \) is \( (-\infty, \infty) \). The range of exponential decay functions is positive real numbers, hence the range is \( (0, \infty) \).

Step 3: Determine the Asymptote

Exponential functions have horizontal asymptotes. For \( f(x) = 10^{-x} \), as \( x \to \infty \), \( f(x) \to 0 \). Thus, the equation of the horizontal asymptote is \( y = 0 \).

Step 4: Analyze Increasing or Decreasing Behavior

The function \( f(x) = 10^{-x} \) is a decreasing function. As \( x \) increases, \( 10^{-x} \) decreases, since larger values of \(-x\) correspond to smaller powers of 10.

Step 5: Graph the Function by Hand

To sketch the graph, note that when \( x = 0 \), \( f(x) = 10^{0} = 1 \). When \( x = 1 \), \( f(x) = 10^{-1} = 0.1 \) and so on. The curve will approach the x-axis (y=0) and decrease from left to right, passing through points like (0, 1) and (1, 0.1).

Step 6: Verify with a Calculator Graph

Use a graphing calculator to input the function \( f(x) = 10^{-x} \). Check that the graph matches your hand-drawn sketch, verifying it decreases from left to right and approaches the horizontal asymptote \( y = 0 \).

Key Concepts

Domain and Range

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 \( x \).

Horizontal Asymptote

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

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.