Problem 46 - Intermediate Algebra

Question

Graph \(f(x)=\log _{4} x\) by reflecting the graph of \(g(x)=4^{x}\) across the line \(y=x\).

Step-by-Step Solution

Answer

Reflect the exponential graph \(g(x) = 4^x\) to graph \(f(x) = \log_{4} x\).

Step 1: Understand the Relationship

The function \(f(x) = \log_{4} x\) is the inverse of \(g(x) = 4^x\). In general, the function \(y = \log_{b} x\) is the inverse of \(y = b^x\), meaning these functions are reflections of each other across the line \(y = x\).

Step 2: Plot the Graph of \(g(x) = 4^x\)

Create a set of points to plot the graph of \(g(x) = 4^x\). Some key points are: \((0, 1)\), \((1, 4)\), and \((2, 16)\). Connect these points with a smooth curve that rises steeply as \(x\) increases.

Step 3: Reflect Points Across \(y = x\)

Take each of the points from \(g(x) = 4^x\) and swap their coordinates to find the points on \(f(x)\). For example, the point \((1, 4)\) on \(g(x)\) becomes \((4, 1)\) on \(f(x)\), and \((0, 1)\) becomes \((1, 0)\).

Step 4: Plot the Graph of \(f(x) = \log_{4} x\)

Using the reflected points, plot the graph of \(f(x) = \log_{4} x\). Key points to plot are: \((1, 0)\), \((4, 1)\), and \((16, 2)\). Draw a smooth curve passing through these points which approaches the y-axis without touching it, forming a logarithmic curve.

Step 5: Verify the Reflection

Ensure that the curve plotted for \(f(x) = \log_{4} x\) passes through the points that are reflections of those from \(g(x) = 4^x\) across the line \(y = x\). This confirms that the inverse relationship and the reflection have been correctly applied.

Key Concepts

Logarithmic Functions

Logarithmic functions are a crucial concept in mathematics, particularly useful in solving equations where the variable is an exponent. The function structure \( f(x) = \log_b x \) denotes a logarithmic function, where \( b \) is the base. This function is essentially the inverse of an exponential function. The base \( b \) must be positive and cannot be one. A logarithmic function answers the question, "To what power must the base be raised to yield the number \( x \)?" For instance, if \( f(x) = \log_{4} x \), it determines the power to which 4 must be raised to produce \( x \).

Logarithmic functions have several vital characteristics:

• They have a vertical asymptote on the y-axis, which means they approach but never touch or cross the y-axis.
• The domain of a logarithmic function is all positive real numbers, \( x > 0 \).
• The range is all real numbers, or \( (-\infty, \infty) \).

Logarithmic functions are used in various fields, from calculating compound interest in finance to expressing the magnitude of earthquakes in geology with the Richter scale.

Exponential Functions

Exponential functions, noted as \( g(x) = b^x \), feature a constant base \( b \) raised to a variable exponent \( x \). Unlike polynomial functions where the base is the variable, here the base remains the same, making these functions grow rapidly.

The choice of base \( b \) plays a key role. Typically, \( b > 1 \) yields growth, while \( 0 < b < 1 \) results in decay. For example, \( g(x) = 4^x \) is an exponential growth function, showing increasing values as \( x \) becomes larger.

Important characteristics of exponential functions include:

• The domain is all real numbers, \( (-\infty, \infty) \).
• The range is all positive real numbers, \( y > 0 \).
• They pass through the point (0,1) since any non-zero number raised to the 0 power equals 1.

Exponential functions are prevalent in describing population growth, radioactive decay, and systems involving compounding.

Graph Reflections

Graph reflections are a method of visualizing inverse functions by flipping them across a line. This process aids in understanding how two functions relate as inverses. In case of the exercise, the graph of the logarithmic function \( f(x) = \log_{4} x \) is the reflection of the exponential function \( g(x) = 4^x \) across the line \( y = x \).

To reflect a graph across the line \( y = x \), you swap the \( x \) and \( y \) coordinates of each point on the function graph. This geometrical transformation helps verify that these reflected points form the inverse function on the graph.

Key steps in graph reflections involve:

• Identifying and plotting key points on the original graph.
• Swapping \( x \) and \( y \) coordinates to find reflected points.
• Sketching the new curve using these reflected points.

Reflecting graphs is an essential skill for visualizing mathematical relationships and understanding the symmetrical properties of inverse functions.