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Chapter 5: Problem 80

Graph the function and its inverse using the same set of axes. Use any method. $$f(x)=\log _{4} x, f^{-1}(x)=4^{x}$$

Short Answer

Expert verified
Graph \( f(x) = \log_{4} x \) and \( f^{-1}(x) = 4^x \) on the same axes, with the line \( y = x \).

Step by step solution

01

- Understand the Functions

We have two functions: the original function is \( f(x) = \log_{4} x \) and the inverse function is \( f^{-1}(x) = 4^x \). The original function takes the logarithm base 4 of \( x \) and the inverse function raises 4 to the power of \( x \).
02

- Determine Key Points for \( f(x) \)

To graph \( f(x) = \log_{4} x \), determine key points by choosing values of \( x \) for which \( f(x) \) is easily found: \( (1, 0), (4, 1), (16, 2) \) since \( \log_{4} 1 = 0 \), \( \log_{4} 4 = 1 \), and \( \log_{4} 16 = 2 \). Outline these points.
03

- Plot \( f(x) \)

On the coordinate system, plot the points: \( (1, 0), (4, 1), (16, 2) \). These support the logarithmic curve in the positive \( x \)-axis direction.
04

- Determine Key Points for \( f^{-1}(x) \)

For \( f^{-1}(x) = 4^x \), choose \( x \) values: \( (-1, \frac{1}{4}), (0, 1), (1, 4), (2, 16) \) since \( 4^{-1} = \frac{1}{4} \), \( 4^0 = 1 \), \( 4^1 = 4 \), and \( 4^2 = 16 \).
05

- Plot \( f^{-1}(x) \)

On the same coordinate system, plot the points: \( (-1, \frac{1}{4}), (0, 1), (1, 4), (2, 16) \). These create the exponential growth curve.
06

- Draw the Line y=x

Draw the line \( y = x \) on the graph. This line aids in visualizing symmetry between \( f(x) \) and \( f^{-1}(x) \). Each point on \( f(x) \) swaps along this line to become a matching point on \( f^{-1}(x) \), confirming they are reflections.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Functions
Logarithmic functions are essential in mathematics and real-world applications. For the function provided, \( f(x) = \log_{4} x \), 'log' represents logarithm, and '4' is the base. A logarithm answers the question: 'To what exponent must the base be raised, to produce a given number?'. In simple terms, \( \log_{4} 16 = 2 \) because raising 4 to the power of 2 gives 16. Understanding this relationship helps in graphing the function accurately. To get neat points on the graph, choose x-values that are easy for calculations:
  • \(\log_{4} 1 = 0\) (since \(4^0 = 1\))

  • \(\log_{4} 4 = 1\) (since \(4^1 = 4\))

  • \(\log_{4} 16 = 2\) (since \(4^2=16\))

These points, \((1, 0), (4, 1), (16, 2)\), will help sketch the logarithmic curve moving rightwards, showing how slowly the function increases.
Exponential Functions
Exponential functions are another cornerstone of mathematics. They often describe rapid growth or decay. For the inverse function given, \( f^{-1}(x) = 4^x \), '4' is the base, and x is the exponent. This function grows quickly, unlike the slow rise of the logarithmic function. Think about how 4 raised to different powers gives us new y-values:
  • \( 4^{-1} = \frac{1}{4} \)

  • \( 4^{0} = 1 \)

  • \( 4^{1} = 4 \)

  • \( 4^{2} = 16 \)

Graphing this will give you points \((-1, \frac{1}{4}), (0, 1), (1, 4), (2, 16)\), helping to visualize the steep upward curve. This shows how each x value leads to a rapidly increasing y value.
Graphing Techniques
Graphing functions requires plotting points and drawing curves that connect them. It helps to start with points you calculated:
  • For \( f(x) = \log_{4} x \): \((1, 0), (4, 1), (16, 2)\)

  • For \( f^{-1}(x) = 4^x \): \((-1, \frac{1}{4}), (0, 1), (1, 4), (2, 16)\)

Plotting these points on the same set of axes shows their curves. Use smooth, continuous lines to connect the dots, making sure to follow their natural progression. For example, the logarithmic function will appear to flatten as it moves right, while the exponential function skyrockets upward dramatically. Also, draw the line \( y = x \), which helps to see the symmetry between a function and its inverse.
Symmetry in Functions
Symmetry is a powerful tool in understanding functions and their inverses. For a function \( f(x) \) and its inverse \( f^{-1}(x) \), the line \( y = x \) acts as a mirror. This line helps to visualize how the graph of \( f(x) = \log_{4} x \) and \( f^{-1}(x) = 4^x \) are reflections of each other. Each point \((a, b)\) on \( f(x) \) corresponds to the point \((b, a)\) on \( f^{-1}(x) \). For example:
  • \((1, 0)\) on \( f(x) = \log_{4} x \) reflects to \((0, 1)\) on \( f^{-1}(x) = 4^x \)

  • \((4, 1)\) on \( f(x) \) reflects to \((1, 4)\) on \( f^{-1}(x) \)

  • \((16, 2)\) corresponds to \((2, 16)\)

Drawing the line \( y = x \) shows these mirrored points visually, aiding comprehension of how inverses work.

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