Problem 82 - A Graphical Approach to College Algebra

Question

Assume that \(f(x)=a^{x},\) where \(a>1\) If \(f\) has an inverse function \(f^{-1},\) sketch \(f\) and \(f^{-1}\) on the same axes.

Step-by-Step Solution

Answer

The sketch includes \(f(x) = a^x\), \(f^{-1}(x) = \log_a(x)\), and the line \(y=x\), showing symmetry.

Step 1: Understand the Function

The given function is an exponential function, defined as \(f(x) = a^x\), where \(a > 1\). Exponential functions are continuous and strictly increasing for \(a > 1\).

Step 2: Identify the Inverse Function

To find the inverse of an exponential function, recall that the inverse of \(f(x) = a^x\) is \(f^{-1}(x) = \log_a(x)\), the logarithmic function to base \(a\). This is because \(a^{x} = y\) implies \(x = \log_a(y)\).

Step 3: Sketch f(x) = a^x

To sketch \(f(x) = a^x\), note that it is an increasing curve starting from the y-intercept \((0,1)\) going upwards as \(x\) increases. As \(x\) goes to negative infinity, \(f(x)\) approaches 0.

Step 4: Sketch f^{-1}(x) = log_a(x)

The graph of \(f^{-1}(x) = \log_a(x)\) is a reflection of \(f(x) = a^x\) over the line \(y = x\). The curve passes through the point \((1,0)\) and is increasing, with the y-axis as a vertical asymptote.

Step 5: Draw the Line y = x

To show the relationship between \(f(x)\) and \(f^{-1}(x)\), draw the line \(y = x\). The functions \(f\) and \(f^{-1}\) should be symmetrical with respect to this line.

Key Concepts

Exponential Functions

Exponential functions are fascinating mathematical expressions. They take the form of \(f(x) = a^x\), where \(a\) is a positive constant greater than 1. These functions have some cool properties: they grow continuously and are strictly increasing as \(x\) increases. This means that as you move along the x-axis to the right, the value of the function gets larger.

One key feature of exponential functions is their y-intercept at the point \((0, 1)\). This is because any number (except zero) raised to the power of 0 is 1. Another interesting aspect is their behavior as \(x\) approaches negative infinity. In this scenario, the function value approaches 0, creating a horizontal asymptote along the x-axis.

Some noteworthy properties include:

• Exponential growth: The function values increase rapidly as \(x\) increases.
• No definite upper limit: Unlike some functions that level off, \(f(x) = a^x\) keeps increasing.
• Simple transformation: Changing \(a\) affects the growth rate but not their fundamental characteristics.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. If you have \(f(x) = a^x\), then its inverse \(f^{-1}(x)\) would be \(\log_a(x)\). Logarithms help you find the power to which you must raise \(a\) to get \(x\). In other words, if \(a^y = x\), then \(y = \log_a(x)\).

Logarithmic functions have special characteristics:

• Passing through the point \((1, 0)\): Since \(a^0 = 1\), this means \(\log_a(1) = 0\).
• Vertical asymptote: They approach the y-axis but do not touch it, making the y-axis a vertical asymptote.
• Slow growth: Unlike exponential growth, logarithms increase slowly.

The beauty of these functions lies in their symmetry with exponential curves. When graphed, a logarithmic function reflects its corresponding exponential curve over the line \(y = x\). This symmetry highlights their inverse relationship and helps to visualize how inverse functions work.

Graphing Techniques

Graphing techniques play a pivotal role in understanding functions and their inverses. For exponential functions, start by identifying key points like the y-intercept at \((0, 1)\). Sketch the curve, ensuring that it rises sharply and approaches 0 as \(x\) moves into negative territory.

When graphing the inverse, the logarithmic function, begin at \((1, 0)\) and draw a curve that rises gently, never quite touching the y-axis. The vertical asymptote (y-axis) is a crucial guide for this plot. Ensure that the curve accurately reflects the exponential one over the line \(y = x\).

Utilize these helpful tips when graphing:

• Draw the line \(y = x\) to examine symmetry between the function and its inverse.
• Mark key points, like where the curve intersects the axes.
• Check the direction of increase or decrease to confirm characteristics.

Mastering these techniques will enable you to efficiently sketch and interpret graphs of both exponential and logarithmic functions.