Problem 90 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The functions \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\) have the same graph.

Step-by-Step Solution

Answer

The statement is TRUE, as the functions \(f(x) = \left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\) are the same and hence have identical graphs.

Step 1: 1. Compare the Functions

The two functions \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\) seem to be quite different at first glance due to the base and exponent present in each function. Let's first find out if these two functions are equivalent by using the Laws of Exponents.

Step 2: 2. Simplify the Second Function Using Laws of Exponents

According to the laws of exponents, if \(a\) is any non-zero real number and \(n\) is an integer, then \(a^{-n} = \frac{1}{a^n}\). Apply this law to function \(g(x)\), \(g(x) = 3^{-x} = \frac{1}{3^x}\)

Step 3: 3. Comparing the Functions Again

After applying the laws of exponents, the two functions are \(f(x) = \left(\frac{1}{3}\right)^{x}\) and \(g(x) = \frac{1}{3^{x}}\). By observing we see these are exactly the same functions and hence have the same graph.

Step 4: 4. Concluding the Result

So the initial statement that the functions \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\) have the same graph is TRUE.

Key Concepts

Laws of Exponents

In mathematics, the laws of exponents are fundamental rules that assist us in simplifying and manipulating exponential expressions. Understanding these laws is crucial when comparing functions like those in the exercise: \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\). Let's explore these laws to uncover how they make the comparison straightforward.

One essential rule to remember is the negative exponent rule, which states:

• \(a^{-n} = \frac{1}{a^n}\)

This means that any base raised to a negative exponent is equivalent to the reciprocal of that base raised to the corresponding positive exponent.

By applying this rule to \(g(x) = 3^{-x}\), we find:
\(g(x) = \frac{1}{3^x}\)

This transformation shows that both functions, \(f(x)\) and \(g(x)\), are equivalent as they simplify to \(\left(\frac{1}{3}\right)^{x}\). Hence, understanding the laws of exponents not only provides deeper insights into the functions but also confirms their equivalence.

Function Equivalence

Function equivalence occurs when two different expressions, although they appear distinct, describe the same function. This concept is vital in analyzing whether two functions represent identical outputs for the same set of inputs.

In our given exercise, we are presented with two functions: \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\). Initially, these expressions look divergent because of the different bases and exponents. However, after applying the laws of exponents, notably the negative exponent rule, \(g(x)\) simplifies to \(\left(\frac{1}{3}\right)^{x}\).

This simplification confirms that both functions are equivalent because they produce the same outputs for any input \(x\). Essentially, function equivalence tells us that despite the differing appearances, these functions will behave identically. Thus, they graph the same curve on the coordinate plane, sharing all characteristics like intercepts, asymptotes, and growth behavior.

Graph of a Function

The graph of a function represents the set of all its possible input-output pairs visually on a coordinate plane. Graphing is a powerful method to see at a glance whether two functions are equivalent, as they will share identical shapes.

For the functions \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\), understanding their graphical representation confirms their equivalence. After using the laws of exponents to simplify \(g(x)\), we see it's the same as the function \(f(x)\).

So, if you were to plot these functions on a graph, they would overlap, tracing the same path. This path would display typical characteristics of an exponential decay due to the base being a fraction less than 1. Key features include:

• A rapid decrease as \(x\) increases.
• A horizontal asymptote along \(y=0\) because the function values approach zero but never quite reach (or cross) it.
• An intercept at \((0, 1)\), since any non-zero number to the power of zero equals one.

Recognizing the equivalence through graphs solidifies understanding, providing a visual affirmation that both functions behave exactly the same in every mathematical sense.