- Improved arithmetic compared to previous lessons. Fewer mistakes. - Getting the hang of using the distance formula between two points. - Successfully found the equations for the inverse of given functions. - Mostly accurate and correct typing of expressions and equations into your CAS calculator.

- Practice working with what you know even when you do not know all the info you want to know for a given question. Often, doing something with the limited info that you do know will generate new info that you can work with further or reshape the info you already have in another way and show you another perspective of that info. - Practice using the formula for the straight distance between two points. - Visualise the reflection of a given function over the line y=x to help you find the domain of the inverse of that function. - Be more careful of typos on your CAS calculator. - Remember the distinction between the inverse and derivative of a function. - Section 2 Q1(b): Sketch the graph with some values of n in mind. Doing this will show you the value of a. - Section 2 Q1(c): It is not always best to think about the specific answer from the very start of answering such a question. Start with the logical basics and go from there. You know that the point P has the coordinates (x,g(x)). And you know that g(x)=0 because P is an x-intercept. Just by that logic you have obtained an equation that you can solve for x. - Section 2 Q2(b): The Null Factor Law through which you make each factor of an expression equal to zero when the entire expression itself equals zero applies not only to polynomials. Your initial working dragged the e^(-x) factor along, but you did not need to drag that factor along. Your revised working rejected that factor from the start via e^(-x)?0, leaving you with a simple polynomial x^2+bx+2=0 on which you could apply b^2-4ac<0 for no solutions. The e^(-x) factor would not give you any solutions anyway.

- Successfully differentiated polynomials. - Successfully used the quadratic formula to solve quadratic equations, albeit with some arithmetic mistakes. - Successfully algebraically rearranged a logarithmic equation into a polynomial equation. (Q3 Tech Free) - Remembered to consider implied constraints when solving logarithmic equations. - Successfully solved a power equation, 5^(2x+1)-2=9×5^x, using a dummy variable substitution. - Successfully calculated the inverse of an exponential function and determined the domain of the inverse function. (Q5 Tech Free) - Successfully solve an inequality, keeping track of the direction of the inequality sign. (Q6 Tech Free) Q7 Tech Free - Successfully found the maximal domain of a logarithmic equation with an unknown real constant. - Successfully found the possible values of the unknown constant given the distance constraint. Note the question was a bit vague as to where the point A was located. You assumed that A was actually the x intercept.

- Be more careful with reading values from your CAS calculator. - Be more careful to not miss an x or t (or whatever the independent variable is) in a polynomial when you are doing working out. - Be careful not to rush your arithmetic. Rushing your arithmetic makes you prone to making mistakes you would not otherwise make, like the previous two mistakes. - Remember that 5^(2x+1)= 5^2x×5 and that 5^2x=(5^x )^2. Simplifying fractions (4?2)/2 ? (4×1.4)/2. I.e., ?2/2 ? 1. The twos here do not cancel out because one of them has a power of 1/2. I.e., one of them is inside a square root sign. Notation - For Q5 of the Tech Free questions you wrote h^(-1) for the inverse function. However, the notation requires you to specify the variable, i.e., h^(-1) (x). You corrected this mistake in the lesson. Well done. Fully answering questions - A question can ask for multiple answers. For example, an inverse equation and its domain. Make sure you take note of what answers are asked of you. You can take note of these before you even start answering the question. (Q5 Tech Free) Solving power equations - Consider the equation 5^x = 5^(3k-3x^2 ). Remember that the goal of solving equation is to find x. So, you can extract the powers and leave out the identical bases, because the only way for the right-hand-side and the left-hand-side to be equal is for the powers to be equal. That is how you get the equation x = 3k-3x^2. Concepts - The discriminant ? =b^2-4ac determines whether a quadratic equation has one, more than one, or no solutions. - The domain of a function generally becomes the range of the corresponding inverse function. (MCQ Q2 Tech Active) Remembering given constraints - In Q7(b) Tech Free you solved for k and wrote k ? 3/5. That is correct, but only partially because the question also said that k is positive. Hence, 0 < k ? 3/5.

- Successfully found the composite functions of pairs of individual functions. The individual functions had various types such as circular, polynomial and exponential. - Correctly identified the range of a transformed circular function such as f(t) = 20 + 5 sin?(nt). (Q1) - Successfully confirmed whether the found composite functions are actually defined by checking whether the range of the inner function is a subset of the domain of the outer function. - Specified logic that would successfully make you find the maximum point in a given composite function. (Q1(c)) - You caught yourself doing C(M) instead of M(C) for Q1. - Successfully substituted given values for variables into a given equation. (Q3(b))

Be more comfortable with working with multiple unknown variables - Q3(c) involved working with multiple unknown variables besides the main independent variable t. Still go ahead with trying to solve the simultaneous equations for t. You can usually still solve the equations by cancelling variables. Connections between math concepts - In a way, functions that can be differentiated using the chain rule are like composite functions in the sense of them having an inner function and an outer function. For example, f(x)=x^2 and g(x)=x+1 can combine into f(g(x))=(x+1)^2. Composite functions can generally be differentiated using the chain rule, like how f(g(x)) here can. Q1(c), 2(c). Arithmetic - Be more careful to not leave behind the square of an expression that is squared. Calculators - Be more careful of whether your calculator is in Degrees mode of Radians mode.

- Had some intuitive idea of what the function y = g(f(x)) is best represented by out of a handful of possible choices of graphs. - Correctly solved for unknown coefficients given two initial functions combined into a known composite function. One of the initial functions had the unknown coefficients. (Example 12)

Algebra - Please be more careful to not miss out squaring a term when substituting a function into another function. (Example 13) -- This mistake can be the difference between getting an incorrect quadratic composite function and the correct quartic (power of 4 polynomial) composite function. -- The purple function below is the same as the green composite function. It shows that we can get the composite function by substituting the inner function for x in the outer function.