Problem 115 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. You can determine the graph of \(f(x)=\log _{6} x\) by graphing \(g(x)=6^{x}\) and reflecting it about the \(x\) -axis.

Step-by-Step Solution

Answer

The statement is false. The graph of \(f(x) = \log_{6}x\) is obtained by reflecting the graph of \(g(x) = 6^x\) about the line \(y=x\), not the \(x\)-axis.

Step 1: Recognize the Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverse of each other. So, if \(f(x) = \log_{6}x\), then its inverse function \(g(x) = 6^x\). In other words, the graph of \(f(x)\) is the reflection of the graph of \(g(x)\) about the line \(y=x\), not the \(x\)-axis.

Step 2: Identify the Reflection Axis

The reflection needed to get the graph of the logarithm from the exponent is around the line \(y=x\), not the \(x\)-axis. The statement given in the exercise suggests reflecting about the \(x\)-axis, which is incorrect.

Step 3: Arrive at Conclusion

From the above discussion, we can conclude that the statement provided in the exercise is indeed false. The actual transformation needed is reflection about the line \(y=x\), not the \(x\)-axis.

Key Concepts

Exponential Functions

Exponential functions are powerful mathematical expressions that involve variables as exponents. A standard form of an exponential function is \( g(x) = b^x \), where \( b \) is a constant base greater than zero, and \( x \) is the exponent. Exponential functions are significant in various real-world applications, such as population growth and radioactive decay.

Some key characteristics of exponential functions include:

• They have a constant rate of growth or decay, meaning the output grows or decreases at a constant multiplicative rate.
• Their graphs are always increasing (if \( b > 1 \)) or decreasing (if \( 0 < b < 1 \)).
• They have a horizontal asymptote, typically the \( x \)-axis, which the graph approaches but never touches.

Consider the function \( g(x) = 6^x \). It grows exponentially as \( x \) increases and has the line \( y = 0 \) as a horizontal asymptote. Understanding exponential functions is a critical step in identifying their inverse relationships with logarithmic functions.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The logarithm function is typically expressed as \( f(x) = \log_b x \), where \( b \) is the base of the logarithm. Logarithmic functions allow us to determine how many times we need to multiply the base to reach a certain value, essentially undoing the effect of exponential growth or decay.

Key features of logarithmic functions include:

• Their domain is restricted to positive real numbers, \( x > 0 \), since we cannot take the logarithm of zero or negative numbers.
• Their range is all real numbers, since the output of a log function can be any real number.
• They pass through the point \((1,0)\), since \( \log_b 1 = 0 \) for any base \( b \).
• The graph of a logarithmic function has a vertical asymptote at \( x = 0 \).

For example, \( f(x) = \log_6 x \) is the inverse of \( g(x) = 6^x \), and hence their graphs are reflections of each other about the line \( y = x \). This important relationship aids in identifying transformations in functions, rather than incorrect assumptions such as reflecting about the \( x \)-axis.

Graph Transformations

Graph transformations involve altering the appearance of a graph. There are several types of transformations, such as translations, reflections, dilations (stretching/shrinking), and rotations. Each transformation alters the graph in a distinct manner.

One common transformation is reflection. For instance, when reflecting the graph of a function over an axis or a line, each point on the graph is mirrored to the opposite side of the axis or line.

In the context of exponential and logarithmic functions:

• Reflection across the line \( y = x \) swaps the \( x \) and \( y \) values, which is why \( g(x) = 6^x \) and its inverse \( f(x) = \log_6 x \) are reflections over the line \( y = x \).
• Reflecting about the \( x \)-axis results in flipping the graph upside down, which is not the transformation needed for creating the graph of a logarithmic function from its corresponding exponential function.

By understanding these transformations, students can better grasp how different types of functions are graphically interconnected, and why certain transformations, like reflection over \( y = x \), properly represents the inverse relationship between exponential and logarithmic functions.