Problem 90 - Algebra and Trigonometry

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 'The functions \(f(x) = (\frac{1}{3})^x\) and \(g(x) = 3^{-x}\) have the same graph' is True.

Step 1: Identify the given functions

We have two functions, \(f(x) = (\frac{1}{3})^x\) and \(g(x) = 3^{-x}\). We should explore whether these two functions are the same.

Step 2: Rewrite g(x) using properties of exponential functions

The exponential function \(g(x) = 3^{-x}\) can be rewritten as \(g(x) = (\frac{1}{3})^x\) using the property that \(a^{-x} = \frac{1}{a^x}\) where \(a\) is any positive number that is not equal to 1.

Step 3: Comparison

Now we can compare the modified function \(g(x) = (\frac{1}{3})^x\) with the original \(f(x) = (\frac{1}{3})^x\). Since they have the same form, it is clear that the graph of both functions will be identical.

Key Concepts

Properties of Exponential Functions

Exponential functions are characterized by their unique properties, which make them distinct from other types of functions. One essential property is their form, usually expressed as \( f(x) = a^x \), where \( a \) is a positive constant known as the base and \( x \) is the exponent. If \( a > 1 \), the function exhibits growth; if \( 0 < a < 1 \), it displays decay. In the exercise, we examined functions with the base less than 1, indicating decay.

Another important property is that the graphs of exponential functions never cross the horizontal axis (x-axis); they asymptotically approach it, but do not touch or cross it. This aligns with the idea that an exponential function for positive base is never zero. More so, the property \( a^{-x} = \frac{1}{a^x} \) is repeatedly beneficial in simplifying and comparing exponential expressions, as well depicted in the step-by-step solution, where \( g(x) = 3^{-x} \) was rewritten to match \( f(x) = (\frac{1}{3})^x \), showcasing that exponential functions with reciprocal bases and opposite exponents are equivalent.

Graphing Exponential Functions

The graph of an exponential function is known as an 'exponential curve'. When graphing these functions, several key features should be noted. The function's base determines the direction of the curve. For instance, if the base is greater than one, the curve will rise as it moves right, and if the base is between zero and one, the curve will fall as it moves right. This behavior exemplifies the exercise's functions, where \( f(x) = (\frac{1}{3})^x \) exhibits a decreasing (decaying) curve.

Identify Key Points

Begin by identifying points through which the curve will pass, specifically where \( x = 0 \), as the function will have a value of 1 regardless of the base because \( a^0 = 1 \). Then, choose a couple of positive and negative values for \( x \) to depict the function's change in direction and approaching nature to the x-axis and y-axis, respectively. Plotting these points on a coordinate plane and connecting them smoothly results in the graph of the exponential function.

Comparing Exponential Functions

When comparing two exponential functions, it’s crucial to analyze both their forms and their bases. Functions with the same base and exponent will yield the same graph, as was seen in the provided exercise where \( f(x) = (\frac{1}{3})^x \) and \( g(x) = 3^{-x} \) were proven to be identical. However, when bases are different, the growth or decay rate changes, resulting in distinct graphs.

Analysis of Growth and Decay

Determining whether the functions are growing or decaying by the formula is key—this is indicated by the base. Decaying functions will have bases between zero and one, while growing functions will have bases greater than one. Once the nature of each function is determined, other attributes such as the rate of growth or decay can be compared by examining the exponents. Higher exponents result in steeper curves for growth and quicker convergence towards zero for decay. These concepts applied correctly can assist in identifying whether exponential functions will overlap, intersect, or diverge from one another as they progress along the graph.