Problem 35 - Calculus of a Single Variable

Question

Verifying Inverse Functions In Exercises 35 and 36 , illustrate that the functions are inverse functions of each other by sketching their graphs on the same set of coordinate axes. $$ \begin{array}{l}{f(x)=4^{x}} \\ {g(x)=\log _{4} x}\end{array} $$

Step-by-Step Solution

Answer

By graphing the functions and verifying their symmetry about the line y=x, it can be determined that the functions \(f(x) = 4^{x}\) and \(g(x) = \log_{4}x\) are indeed inverses of each other.

Step 1: Graph the functions

The first step is to graph the functions on the same set of axes. In doing so, you should take care to plot the functions accurately to represent how they change with respect to each other. You may find it helpful to use a graphing calculator or similar tool. Plot \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\).

Step 2: Draw the line y = x

The second step is to graph the line y = x on the same set of axes. This line will serve as the mirror line, which will be used to verify whether the two functions are inverses of each other.

Step 3: Evaluate the symmetry

The final step is to evaluate the symmetry of the graphs of the functions. The graphs of the functions and the line y = x should all share the same point of intersection. From this point, the graphs of the functions should reflect each other along the line y = x. If the graphs of the functions are symmetrical to the line y = x, then the functions are inverses of each other.

Key Concepts

Graphing Functions

Graphing functions is an essential skill in mathematics, allowing students to visualize relationships between variables. When graphing the function f(x) = 4^x, it's important to recognize that this represents an exponential function. It will start off slowly increasing and then rapidly rise as x becomes larger. On the other hand, its inverse, g(x) = log_4(x), is a logarithmic function, which mirrors the exponential's behavior in a reciprocal manner: it rises quickly at first and then levels off as x increases.

To graph these functions accurately, it's useful to plot key points and observe the general shape of each curve. For the exponential function, ensure to include points where x is zero and negative to show the rapid decline towards zero. With the logarithmic function, highlighting where it crosses the x-axis at x=1 (since any log base itself is 1) helps in understanding its progression. Graphing tools or software can aid in plotting these more complicated shapes, but a basic sketch will still demonstrate the inverse nature when compared with its counterpart plotted on the same axes.

Exponential Functions

Exponential functions, such as f(x) = 4^x, have unique characteristics that set them apart from other functions. The base, in this case 4, is raised to the power of x, which serves as the independent variable. As x increases, the output of the function grows exponentially, which in graphical terms creates a J-shaped curve. One key property to remember is that exponential functions always pass through the point (0, 1), since any non-zero number to the power of zero equals one.

Understanding this exponential growth is crucial, as it appears in various real-world contexts such as compound interest, population growth, and radioactive decay. When dealing with such functions, be aware that they are not symmetrical; they don't reflect across the y-axis, but they do have a horizontal asymptote, usually the x-axis, which the function approaches but never touches for negative values of x.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and have their own distinct shape and properties. For the given function g(x) = log_4(x), it's important to recognize that it rises quickly and then begins to flatten out as the input increases; this is the reverse of how the exponential function behaves. A logarithmic function will always cross the x-axis at x=1, no matter the base, since the logarithm of 1 is always zero.

The concept of logarithms can be challenging, but remember, they answer the question: 'To what power must the base be raised to obtain the given number?' For example, log_4(16) = 2 because 4 squared is 16. Logarithmic functions often come up in scientific data involving magnitudes, such as the Richter scale for earthquakes or pH level in chemistry, indicating their practical importance in interpreting exponential changes.

Symmetry

Symmetry in mathematics can be a powerful tool for understanding relationships between functions, especially when evaluating inverse functions. When graphing an exponential function and its logarithmic inverse, their symmetry across the line y = x illustrates that one function undoes the action of the other. In the case of f(x) = 4^x and g(x) = log_4(x), you should notice that for any point on the graph of one function, its reflection across the line y = x is a corresponding point on the graph of the other function.

This symmetry is not only visually compelling but also proves mathematically that one function is indeed the inverse of the other. This concept is further validated when you consider that substituting one function into the other returns the original input, which is the hallmark of inverse operations: f(g(x)) = x and g(f(x)) = x. Keeping an eye out for symmetry is a useful check that you have correctly understood and graphed the functions.