Welcome back to Professor Baker’s Math Class! In Chapter 1-5, Section 1, Lesson 4, we are bridging the gap between algebraic manipulation and transcendental functions. This session focuses heavily on the concept of "undoing" operations—first through inverse functions, and then by exploring the inverse relationship between exponentials and logarithms.

1. Inverse Functions

We begin by revisiting the concept of inverses. Remember, an inverse function essentially reverses the input and output of the original function. Geometrically, this results in a reflection across the line $y = x$. In this lesson, we tackle three specific types:

  • Quadratic and Square Root: We explore how restricting the domain of a quadratic function (e.g., $x \ge 0$) allows us to find its inverse, which is a square root function.
  • Rational Functions: We practice the algebraic steps to solve for the inverse, $f^{-1}(x)$, involving rational expressions.

2. Exponential Functions: Graphs and Applications

Next, we move into exponential growth and decay. An exponential function typically takes the form $f(x) = a(b)^x$. Understanding these functions is crucial for modeling real-world scenarios like population growth or radioactive decay.

Key concepts covered include:

  • Asymptotes: Identifying the horizontal asymptote. For a function $f(x) = b^x + c$, the asymptote is the line $y = c$.
  • The Natural Base $e$: We introduce Euler's number, $e \approx 2.718$, and graph functions such as $f(x) = a(e)^{x-b} + c$.
  • Domain and Range: We determine that the domain of a standard exponential function is $(-\infty, \infty)$, while the range is determined by the horizontal asymptote.

3. Logarithmic Functions

Logarithms are the inverses of exponential functions. If you can handle exponentials, you can handle logs! We focus on converting between the two forms using the definition:

$$y = \log_b(x) \iff b^y = x$$

This section includes:

  • Graphing: Visualizing logarithmic functions and identifying their vertical asymptotes (where the argument of the log equals zero).
  • Transformations: Just like with other functions, we look at how to shift the graph of a log function vertically and horizontally.
  • Natural Logarithms: Converting between natural log ($\ln$) and exponential equations involving $e$.

4. Solving Equations

Finally, we apply these concepts to solve equations. We look at exponential equations where we can find a common base. For example, to solve $4^x = 8$, we rewrite both sides with base 2: $(2^2)^x = 2^3$, leading to $2x = 3$.

By the end of this lesson, you should feel comfortable navigating between exponential and logarithmic forms and interpreting their graphs. Keep practicing those transformations!