Welcome to Week 12! We are moving into Chapter 12, covering Section 12-1 and Section 12-2. This week is pivotal as we transition from standard algebraic operations into function composition and the exciting world of exponential growth and decay.

Section 12-1: Composite Functions

In this section, we look at functions within functions. A composite function is created when one function is substituted into another. The notation used is $(f \circ g)(x)$, which is read as "$f$ composed with $g$" or "$f$ of $g$ of $x$."

Mathematically, this means:

$$(f \circ g)(x) = f(g(x))$$

Example from Class Notes:
If we have $f(x) = 3x$ and $g(x) = 1 + x^2$, to find $(f \circ g)(x)$, we substitute the entire function $g(x)$ into every instance of $x$ in $f(x)$:

  • $f(g(x)) = 3(1 + x^2)$
  • $(f \circ g)(x) = 3 + 3x^2$

Remember, the order matters! $(f \circ g)(x)$ is usually not the same as $(g \circ f)(x)$.

Section 12-2: Exponential Functions

We are also introducing exponential functions, which differ significantly from the linear functions we are used to. As we discussed with the "Penny Example," exponential growth starts slow but accelerates rapidly.

The standard form is $f(x) = a^x$. Per the rules outlined in your notes:

  • The base $a$ can never be negative.
  • Exponential Growth: When $a > 1$ (e.g., $y = 2^x$).
  • Exponential Decay: When $0 < a < 1$ (e.g., $y = (\frac{1}{2})^x$).

We also applied this to real-world scenarios using the Compound Interest Formula:

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

Where $P$ is the principal amount, $r$ is the rate, $n$ is the number of times compounded, and $t$ is time.

Important Announcement: Thanksgiving Grade Recovery

I have noticed several students are missing quizzes, which is significantly affecting grades. In honor of Thanksgiving, I am offering one last shot to make up any missed quiz for half credit.

  • If you are unsure which quizzes you are missing, you must email me to find out.
  • Deadline: All make-up quizzes must be completed by 6:00 AM on November 23rd.
  • I will be posting the answers to all quizzes on Friday to help you study for the final exam, so no late submissions will be accepted after the deadline.

Weekly Assignments & Resources

Please review the attached class notes (PDFs) for detailed step-by-step examples of quadratic reviews and function composition.

  • Homework 12-1: Pages 862-863, #9-20 and #31-40
  • Homework 12-2: Page 874 (Graphs #11-39 odd) and Page 875 (#54-60)

Good luck with the new material and the grade recovery opportunity!