Sections 11 and 12

So so...
I would have liked to see more limits with square roots of the form \( 0/0 \), and limits that do not exist, but the ones you presented are ok.
I am also missing some nice exponential and logarithmic equations—they will be an important part of this portion of the final exam.
None of the problems involved sign charts. Do not forget about those!
Another thing I am missing is shifts, stretching, compression, mirror images...
The problem on composition is good, but I would have liked to see the computation of some domains of compositions as well.
There was a repeated topic, and one question that did not make any sense. I dismissed those two questions.

1. Find the domain of the function
   \begin{equation} f(x) = \frac{3x-7}{4x-5} \end{equation}
2. Given \( f(x) = x^2-5, g(x) = 7x+15 \), find \( f(g(x)), f(x)/g(x), f(f(x)), g(g(x)) \), and \( f(x)-g(x) \).
3. Evaluate \( \log_2 80 - \log_2 5 \).
4. Compute the following limits:
   \begin{align} &\lim_{x\to -2} \frac{x+2}{x^3+8} \\ &\lim_{h\to 0} \frac{(7+h)^2-49}{h} \\ &\lim_{x\to -2^+} \frac{x+1}{x+2} \\ &\lim_{x\to 2} \sqrt[3]{x^5 -3x^2 +7} \end{align}
5. For what values of the constant \( C \) is the function \( f(x) \) continuous in \( (-\infty, \infty) \)?
   \begin{equation} f(x) = \begin{cases} Cx^2 +2x &\text{ if } x<2 \\ x^3-Cx &\text{ if } x\geq 2 \end{cases} \end{equation}

Sections 15 and 16

This is a much better selection of limits and a far more challenging selection of domains. I like that you also tried some intuitive ranges. Good for you!
Your logarithmic equations are evil: I did enjoy that very much.
I thoroughly enjoyed the diversity of problems, and the careful choices of functions, to avoid repetition. Amazing team work. Expect a bunch of extra credit for this exam.

1. Find domain and range of the following functions:
   \begin{align} f(x) &= -\sqrt{x} - 2 \\ f(x) &= -\sqrt{49-x^2} \\ f(x) &= \frac{5x+1}{\sqrt{x^2+5x+6}} \end{align}
2. Compute the following limits:
   \begin{align} &\lim_{x\to 1^+} \frac{x-1}{x^2-1} \\ &\lim_{x\to 2} \frac{x^2+x-6}{x-2} \\ &\lim_{x\to 0^-} \frac{\sqrt{x+1}-1}{x} \\ &\lim_{x\to 3} -\frac{19}{3-x} \\ &\lim_{x\to -1} \frac{2x^2+2}{x^2+4x-4} \\ &\lim_{x\to 1} \bigg( \frac{2+4x}{4+11x^2+3x^4} \bigg)^3 \end{align}
3. Solve for \( x \) in the equation \( \log_4 (x+1) + \log_{16}(x+1) = \log_4 8 \),
4. Simplify \( \log_7 2401 \).
5. Expand the following expression:
   \begin{equation} \log_3 \bigg( \frac{9x^{11}}{\sqrt{y}} \bigg) \end{equation}
6. Find the value of the constant \( n \) that will make the function \( f(x) \) continuous everywhere.
   \begin{equation} f(x) = \begin{cases} \frac{x^2-1}{x-1} &\text{ if } x<1 \\ nx-2 &\text{ if } x\geq 1 \end{cases} \end{equation}
7. Find \( f \circ g \) and its domain, if
   • \( f(x) = \sqrt{x}, g(x) = x-4 \).
   • \( f(x) = 3x^2-4x+7, g(x) = \sqrt{x- \tfrac{2}{x}} \).
   • \( f(x) = \dfrac{1}{x+2}, g(x) = \dfrac{3}{x-3} \).
   • \( f(x) = \dfrac{x}{x-1}, g(x) = \sqrt{x} \).
8. Simplify the following expressions
   • \( 8^{4x} 8^{2000x} \).
   • \( e^{12x^2} e^{-73x} \).
   • \( e^{x^2} e^{-8x} e \).
9. Find an exponential function of the form \( f(x) = Ca^x \), if we know that its graph contains the points \( (3,3) \) and \( (9,30) \).
10. What is the effect of the following operations on the graph of \( f(x) \)?
    • \( 4f(x) \).
    • \( -f(6x) \).
    • \( 5 f(x-3)-4\).