91Ó°ÊÓ

An exponential function, like the one given in the exercise, is a function of the form \( f(x) = e^x \). In this type of function, the variable \( x \) is an exponent. Let's break down some key aspects of exponential functions:

Exponential Growth: The function \( f(x) = e^x \) represents exponential growth. As \( x \) increases, \( f(x) \) increases rapidly. Conversely, as \( x \) decreases, \( f(x) \) approaches 0 but never quite reaches it. This behavior is why exponential functions are used to model situations involving rapid growth or decay, such as population growth or radioactive decay.

Key Points: In the context of the provided exercise, the key points \((0,1)\), \((1,e)\), and \((-1,1/e)\) help us draw the curve accurately. These points show that the function always produces positive results and never crosses the x-axis. The curve of the exponential function increases steeply for positive values of \( x \) and flattens as it moves towards negative values of \( x \).

Logarithmic functions are the inverses of exponential functions. In the given exercise, the inverse function of \( f(x) = e^x \) is \( f^{-1}(x) = \ln x \). Here's what you need to know about logarithmic functions:

Logarithmic Growth: A logarithmic function grows slowly compared to the exponential function. For values of \( x \) greater than 1, \( \ln x \) increases, but at a decreasing rate. For values between 0 and 1, the function produces negative results, showing that \( \ln x \) can dip below the x-axis.

Key Points: To understand the function's behavior, plotting points like \((1,0)\), \(( e, 1 )\), and \((1/e, -1)\) is crucial. These points tell us that the logarithmic function crosses the x-axis at \( x = 1 \) and the y-axis at \( y = 0 \), and moves in the opposite direction of the exponential function relative to the line \( y = x \). The logarithmic function increases slowly and infinitely as \( x \) grows larger.

Symmetry is a crucial concept when dealing with functions and their inverses. Let's explore what symmetry means in this context:

Line of Symmetry: When graphing a function and its inverse, such as \( f(x) = e^x \) and \( f^{-1}(x) = \ln x \), the graphs will be symmetrical about the line \( y = x \). This line acts as a mirror, reflecting one graph onto the other. If you fold your graph paper along the line \( y = x \), the function and its inverse would overlap perfectly.

Visual Confirmation: By drawing the line \( y = x \) along with the curves of the functions, you can visually confirm their symmetry. For the given functions, the exponential curve \( f(x) = e^x \) rises rapidly while the logarithmic curve \( f^{-1}(x) = \ln x \) rises slowly. Their points of intersection, such as \((1,1)\), are on the line \( y = x \), further verifying they are reflections of each other. This symmetry property is true for all function-inverse pairs.