Verified Answer :

  • Exponential function \(f(x) = 3^x\) and logarithmic function \(f^{-1}(x) = \log_3 x\) are inverse functions reflecting over \(y = x\).
  • Key points for \(f(x)\) include \((-1, 1/3), (0, 1), (1, 3)\).
  • Corresponding points for \(f^{-1}(x)\) are \((1/3, -1), (1, 0), (3, 1)\).
  • Plotting these on the same Cartesian plane demonstrates their symmetric relationship, with \(y=0\) as the horizontal asymptote for \(f(x)\) and \(x=0\) as the vertical asymptote for \(f^{-1}(x)\).
  • The final representation is a visualization showing both curves reflecting across the line \(y = x\).

Explanation

  1. Analyzing the Functions and Their Relationship

To graph \(f(x) = 3^x\) and its inverse \(f^{-1}(x) = \log_3 x\), we first recognize that these are exponential and logarithmic functions with the same base \(b=3\). The inverse relationship means that the graphs are reflections of each other across the identity line \(y = x\). The domain of \(f(x)\) is all real numbers \((-\infty, \infty)\) with a range of \((0, \infty)\), while the domain of \(f^{-1}(x)\) is \((0, \infty)\) with a range of \((-\infty, \infty)\).

  1. Finding Key Points for f(x) = 3^x

We identify key points on the graph of \(f(x) = 3^x\) to plot:

  • For \(x = -1\), \(f(-1) = 3^{-1} = 1/3\). Point: \((-1, 1/3)\)
  • For \(x = 0\), \(f(0) = 3^0 = 1\). Point: \((0, 1)\)
  • For \(x = 1\), \(f(1) = 3^1 = 3\). Point: \((1, 3)\)

The graph of \(f(x)\) has a horizontal asymptote at \(y = 0\) as \(x \to -\infty\).

  1. Finding Key Points for f^{-1}(x) = log_3 x

Since \(f^{-1}(x)\) is the inverse of \(f(x)\), we obtain its points by swapping the \(x\) and \(y\) coordinates of the points identified for \(f(x)\):

  • From \((-1, 1/3)\), we get \((1/3, -1)\).
  • From \((0, 1)\), we get \((1, 0)\).
  • From \((1, 3)\), we get \((3, 1)\).

The graph of \(f^{-1}(x)\) has a vertical asymptote at \(x = 0\) as \(x \to 0^+\).

  1. Graphing the Functions and Verifying Symmetry

To graph both functions on the same set of axes:

  1. Plot the line \(y = x\) as a dashed line to visualize the symmetry.
  2. Plot the points \((-1, 1/3), (0, 1), (1, 3)\) and draw the smooth curve for \(f(x) = 3^x\), approaching the x-axis as \(x\) decreases.
  3. Plot the points \((1/3, -1), (1, 0), (3, 1)\) and draw the smooth curve for \(f^{-1}(x) = \log_3 x\), approaching the y-axis as \(x\) approaches 0 from the right.
  4. Both curves exhibit the property of symmetry across the line \(y = x\).

Examples

The concept of inverse functions, such as the exponential and logarithmic relationship, is fundamental in modeling processes that grow or decay over time. For example, if you are tracking the growth of bacteria in a lab (\(f(x) = 3^x\)), knowing the inverse allows you to calculate the time required to reach a specific population size (\(f^{-1}(x) = \log_3 x\)). This relationship is widely used in finance to solve for time in compound interest problems and in science to measure acidity using the pH scale.

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