Problem 46 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph \(f(x)=\left(\frac{1}{4}\right)^{x}\) and \(g(x)=\log _{\frac{1}{4}} x\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graph of \(f(x) = (\frac{1}{4})^{x}\) starts from the point \(x = -1, y = 4\), passes through \(x = 0, y = 1\) and approaches the x-axis as \(x\) increases. The graph of \(g(x) = \log _{\frac{1}{4}} x\) is the reflection of the exponential function's graph over the line \(y = x\). It starts at \(x = 4, y = -1\), passes through \(x = 1, y = 0\) and then approaches the y-axis as \(x\) decreases.

Step 1: Graph the Exponential Function

Visualizing an exponential function requires plotting points and connecting them. Start with some simple points. When \(x = 0\), \(f(x) = (\frac{1}{4})^0 = 1\). When \(x = 1\), \(f(x) = (\frac{1}{4})^1 = \frac{1}{4}\). When \(x = -1\), \(f(x) = (\frac{1}{4})^{-1} = 4\). Using these points and the nature of exponential function, draw the curve of the function \(f(x) = (\frac{1}{4})^x\).

Step 2: Graph the Logarithmic Function

Now, graph \(g(x) = \log _{\frac{1}{4}} x\). Note that the graph of a logarithmic function is the inverse of the exponential function with the same base if the base is greater than 1. However, since our base here is less than 1, the graph will be a reflection of the exponential function's graph over the line \(y = x\). Begin by swapping all \(x\) and \(y\) values from the exponential function's points. For example, when \(x = 4\), \(y = -1\), and when \(x = 1\), \(y = 0\). Using these points, draw the curve of the function \(g(x) = \log _{\frac{1}{4}} x\).

Step 3: Combine Both Graphs

Once both graphs are ready, combine them on the same rectangular coordinate system.

Key Concepts

Exponential Function Graph

When we talk about graphing an exponential function like
\(f(x)=(\frac{1}{4})^{x}\), it involves plotting a set of ordered pairs and understanding how the function behaves. A classic characteristic of the exponential function is that it changes rapidly. If the base of the exponent, in this case, \(\frac{1}{4}\), is less than one, the graph will show a rapid decrease as the value of \(x\) increases. This is because any number between zero and one raised to a higher power keeps getting smaller.

To graph \(f(x)=(\frac{1}{4})^{x}\), we start by choosing values for \(x\) and computing the corresponding \(y\) values, or \(f(x)\). For example:

• At \(x = 0, f(x) = (\frac{1}{4})^0 = 1\)
• At \(x = 1, f(x) = (\frac{1}{4})^1 = \frac{1}{4}\)
• At \(x = -1, f(x) = (\frac{1}{4})^{-1} = 4\)

Plotting these points, we get a curve that starts high when \(x\) is negative and approaches the \(x\)-axis without ever touching it as \(x\) goes to positive infinity. The \(y\)-value at \(x = 0\) is a key point known as the graph's \(y\)-intercept.

Logarithmic Function Graph

Graphing a logarithmic function like \(g(x)=\log _{\frac{1}{4}} x\) also involves plotting points, but the behavior of the graph is quite distinct from its exponential counterpart. Logarithmic functions are inversely related to exponential functions. This means that if we were to take the exponential function graph and flip it over the line \(y = x\), we would get the logarithmic function graph.

For the function \(g(x)=\log _{\frac{1}{4}} x\), since our base \(\frac{1}{4}\) is less than one, the graph will decline as we move along the positive \(x\)-axis. It is important to switch the \(x\) and \(y\) coordinates from the exponential function to find points for the logarithmic function. For instance:

• When \(y = 4\), \(x = -1\) becomes \(x = 4, y = -1\) in the logarithmic graph
• When \(y = 1\), \(x = 0\) remains \(x = 1, y = 0\) since they intersect at the point of inversion

The shape of the logarithmic graph will be a mirror image of the exponential graph across the line \(y = x\), with the curve approaching the \(y\)-axis near zero, but never actually reaching it, signifying that the logarithm of zero is undefined.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane formed by the intersection of a horizontal line called the \(x\)-axis and a vertical line called the \(y\)-axis. This system is fundamental for graphing functions, as it allows us to plot points defined by pairs of numbers, which are the values of \(x\) (horizontal position) and \(y\) (vertical position).

To graph any function, we calculate coordinates, or pairs of \((x, y)\), and place them in this plane. For example, in our given functions, the point \((0,1)\) refers to the \(y\)-intercept of the exponential function, and the point \((1,0)\) corresponds to the \(x\)-intercept of the logarithmic function. When we combine graphs of different functions in a single coordinate system, as with \(f(x)\) and \(g(x)\), we can easily compare their properties and see how they relate to each other, such as being inverses.

Inverse Functions

Inverse functions essentially reverse the roles of inputs and outputs of the original function. If you have a function \(f(x)\), and you can find a function \(g(x)\) such that for every \(x\) in the domain of \(f\), the equation \(f(g(x)) = x\) holds true, then \(g\) is the inverse of \(f\), denoted as \(f^{-1}(x)\). When graphed, the points of \(f(x)\) and \(g(x)\) reflect across the line \(y = x\), because their \(x\) and \(y\) coordinates are swapped.

In the context of exponential and logarithmic functions, these two types of functions are natural inverses of each other. This is why the graph of our logarithmic function \(g(x)\) appears as a reflection of the exponential function \(f(x)\) over the line \(y = x\), signifying this deep relationship between exponentiation and logarithms, where one undoes what the other does.