Problem 29 - A Graphical Approach to Precalculus with Limits

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. $$f(x)=4^{-x}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\); Range: \((0, \infty)\); Asymptote: \(y = 0\); Decreasing function.

Step 1: Identify the Function Type

The given function is \( f(x) = 4^{-x} \). This is an exponential function where the base is 4 and the exponent is \(-x\). Exponential functions typically involve continuous growth or decay.

Step 2: Determine the Domain and Range

The domain of an exponential function is all real numbers, \( (-\infty, \infty) \). The range of a decreasing exponential function \( f(x) = 4^{-x} \) is \( (0, \infty) \), since \( 4^{-x} \) can never be zero or negative.

Step 3: Identify and Plot Intercepts

To find the y-intercept, calculate \( f(0) = 4^0 = 1 \). Therefore, the y-intercept is at \((0, 1)\). There is no x-intercept for this function because 4 to any negative power cannot equal zero.

Step 4: Determine the Asymptote

For exponential decay functions like \( f(x) = 4^{-x} \), the horizontal asymptote is the line \( y = 0 \). This is because as \( x \) becomes increasingly positive, \( 4^{-x} \) approaches zero but never actually reaches it.

Step 5: Analyze the Behavior of the Function

The function \( f(x) = 4^{-x} \) is a decreasing function. As \( x \) increases, the function's value decreases because the exponent \( -x \) means the base 4 is raised to a more negative power, resulting in smaller numbers.

Step 6: Create Graph using a Calculator

Using a graphing calculator, plot \( f(x) = 4^{-x} \) to visually verify the characteristics identified: the y-intercept at \((0, 1)\), no x-intercept, the horizontal asymptote at \( y = 0 \), and the decreasing nature of the function.

Key Concepts

Domain and Range

When dealing with exponential functions, like our function \( f(x) = 4^{-x} \), the domain represents all the possible input values \( x \) can take. For exponential functions, the domain is all real numbers, which means \( x \) can be any number from \(-\infty\) to \(\infty\). This gives the function a wide breadth of input possibilities.

However, the output, or the range, is what the functions can produce. In the case of \( f(x) = 4^{-x} \), the range is limited to positive values because no matter what negative exponent we use in the base 4, we never get a negative or zero when raised to such powers. Hence, the range is \( (0,\infty) \), which means \( f(x) \) is strictly positive for all real inputs. This reflects the core behavior of exponential decay, where the function values get smaller but never reach zero.

Asymptotes

An asymptote in a graph is a line that the graph of the function gets infinitely close to but never touches. In the graph of \( f(x) = 4^{-x} \), this behavior is evident as it approaches a horizontal line as \( x \) increases.

For the mentioned function, there is a horizontal asymptote at \( y = 0 \). This occurs because as \( x \) becomes a large positive number, the value of \( 4^{-x} \) shrinks towards zero. However, exponential functions involving negative exponents never actually reach zero, just coming extraordinarily close. This unique feature is characteristic of exponential decay functions, which prominently seem to "hug" the x-axis without crossing it.

Decreasing Function Analysis

In the analysis of exponential functions, understanding if a function is increasing or decreasing is key. The function \( f(x) = 4^{-x} \) demonstrates what is known as exponential decay. This means it is a decreasing function.

As you increase \( x \), since the exponent is negative, \( 4^{-x} \) grows smaller because the base is multiplied by reciprocal powers of four. Specifically, moving towards more positive values of \( x \) decreases \( f(x) \) because you are effectively raising the base to larger negative exponents, which results in smaller numbers. Thus, as \( x \) goes on to positive infinity, \( f(x) \) keeps decreasing, highlighting the characteristic decay of exponential functions.