Let's start this off with a quick review of the main concepts learned from Grade 11 Functions.

First off, a function is defined as a relationship/expression where there is **no more than one** value of the dependent variable associated with every unique value of the independent variable.

In simpler terms, for the independent variable given by \(x\) and function given by \(f(x)\) there is only 1 \(f(a)\) for every \(a\) in \(x\). The simplest example of a function is given by: \(f(x) = x\).

Due to this rule, all functions must pass the vertical line test. This is where if you were to pass a vertical line through the entirety of that function, so for \(x \in (-\infty,\infty)\), there will always be at most one point of contact between the vertical line and the graphed function of interest.

The functions that we covered previously include the following:

\(\begin{align} f_1(x) & = x \\ f_2(x) & = x^2 \\ f_3(x) & = x^3 \\ f_4(x) & = \vert x \vert \\ f_5(x) & = \frac{1}{x} \\ f_6(x) & = \sin(x) \\ f_7(x) & = \cos(x) \\ f_8(x) & = \tan(x) \\ \end{align}\)

Another property of functions that was covered was the concept of **transformations**. Any basic function, \(f(x)\), has an associated family of functions, \(g_1(x),g_2(x), ...,g_n(x)\), all of which could be created through transformations.

Transformations follow the general formula: \(g(x) = a*f(k*(x-h))+d\), for \(a,k,h,d \in \rm I\!R\). These constants control their own aspects of the transformations, which we've covered in Lesson 5: Transformations of Functions.

The general rule to these is that the constants outside of \(f(x)\), \(a\) and \(d\), control the **vertical** aspects of transformations whereas the parameters inside, like \(k\) and \(h\), will control the **horizontal** aspects. In addition, multiplied constants, \(a\) and \(k\), will stretch/compress the function and those added/subtracted, \(h\) and \(d\), will shift the function, each in their own respective directions. Hence, \(a\) will vertically expand/compress the function, \(k\) will horizontally compress/expand the function, \(h\) will shift/translate the function horizontally, and \(d\) will vertically shift/translate the function.

The next key aspect of this review is the domain and range of functions and how they are affected by these transformations. The domain describes the x-values along which the function exists. The range describes the set of y-values along which the function exists. For the basic functions that were described above, the domain and range for each are as follows:

\(\begin{align} f_1(x) & = x \\ & D_1 = \{ x \in \rm I\!R \} \\ & R_1 = \{ y \in \rm I\!R \}\\ f_2(x) & = x^2 \\ & D_2 = \{ x \in \rm I\!R \} \\ & R_2 = \{ y \in \rm I\!R \vert y \geq 0 \}\\ f_3(x) & = x^3 \\ & D_3 = \{ x \in \rm I\!R \} \\ & R_3 = \{ y \in \rm I\!R \}\\ f_4(x) & = \vert x \vert \\ & D_4 = \{ x \in \rm I\!R \} \\ & R_4 = \{ y \in \rm I\!R \vert y\geq 0 \}\\ f_5(x) & = \frac{1}{x} \\ & D_5 = \{ x \in \rm I\!R \vert x \neq 0 \} \\ & R_5 = \{ y \in \rm I\!R \vert y \neq 0 \}\\ f_6(x) & = \sin(x) \\ & D_6 = \{ x \in \rm I\!R \} \\ & R_6 = \{ y \in \rm I\!R \vert -1 \leq y \leq 1 \}\\ f_7(x) & = \cos(x) \\ & D_7 = \{ x \in \rm I\!R \} \\ & R_7 = \{ y \in \rm I\!R \vert -1 \leq y \leq 1 \}\\ f_8(x) & = \tan(x) \\ & D_8 = \{ x \in \rm I\!R \vert x \neq \frac{\pi}{2} + \pi n; n \in \mathbb{Z}\} \\ & R_8 = \{ y \in \rm I\!R \}\\ \end{align}\)

Both the domain and range of functions can change when the function is transformed. So the domains and ranges listed above are true for the base function they correspond to, but not necessarily for the family of functions that can be generated from that base function. We will mainly just focus on the domain and how it changes for the remainder of this course but it is important to remember that the rules for the range may also change depending on the transformation applied.

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