Hints

You can find hints for some of the exercises in the idividual tabs.

Geogebra

Use the algebra tool to define functions you want to plot

Use the grpah tools to place points, find intersects and extrema of the functions

You can change the scale of the axis (zoom in or out) by holding Shift on your keyboard and dragging the axis using the mouse.

E2

Recall the definition of the absolute value \[f(x) = |x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases} \] Use the equation above to evaluate the absolute value for \(f(x-2) = |x-2|\). Then solve the inequality and do the plot.

We indicate by \(x \in (a, b)\) that \(x\) is in the interval with the boundary points \(a\), \(b\). We can also write this as \(a<x<b\).

Side comment about interval notation. Note the use of ( and [ for open and closed interval

• \(x \in [a, b)\) is equivalent to \(a \leq x < b\)
• \(x \in (a, b]\) is equivalent to \(a < x \leq b\)
• \(x \in [a, b]\) is equivalent to \(a \leq x \leq b\)
• \(x \in (-\infty, \infty)\) is equivalent to \(-\infty < x < \infty\) (for \(\infty\) we use always onpen intervals)

E3

Solving for \(x\) the equation \(f(x)=g(x)\) means finding the value of \(x\). For example, for \(f(x) = 3x+2, g(x)=x/2-3\) we have \[\begin{eqnarray} f(x) & = & g(x) \\ 3x+2 & = & x/2 - 3 \\ 3x - x/2 & = & -3 - 2 \\ 6x - x & = & 2(-3 - 2) \\ 5x & = & -10 \\ x & = & -2 \end{eqnarray}\]

Recall the definition of the absolute value \[f(x) = |x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases} \] to find the values such as \(f(5-2x) = |5-2x|\).

E4 & E5

You can start from the relation of the angle \(180^{o} = \pi \text{ radians}\) therefore for any angle \(\theta\) we have the equality \[\frac{\theta^{o}}{180} = \frac{\theta \text{ rads}}{\pi}\] and hence \[\begin{eqnarray} \theta^{o} & = & \frac{180}{\pi} \theta \text{ rads}\\ \theta \text{ rads} & = & \frac{\pi}{180} \theta^{o} \end{eqnarray}\]

E8

We use the unit circle for obtaining the values of the \(\sin\) and \(\cos\) function. Think about the circle below. What are the values of A, O and H and why?

Extra hint: remember pythagoras theorem?

E11

To find an inverse function \(f^{-1}\) of the function \(f\)

1. Solve the equation \(y = f(x)\) for \(x\).
2. Interchange the variables \(x\) and \(y\) and write \(y=f^{-1}(x)\)

Eg. \(f(x) = 3x-4\)

1. \(y = 3x - 4 \Rightarrow 3x = y+4 \Rightarrow x = \frac{1}{3}y + \frac{4}{3}\)
2. \(y = f^{-1}(x) = \frac{1}{3}x + \frac{4}{3}\)

E14

Recall that exponential and logarithm functions are inverses of each other so that \[\log_b x = y \text{ if and only if } b^y = x \enspace .\]

Further recall the property of inverse functions \[f^{-1}\big( f(x)\big) = x \enspace .\] You may also need to use the properties of exponents and logarithms listed in sections 2.6.1.1 and 2.6.2.2 here.

E20

The joint probability distribution of independent random variables is the product of the marginal distributions.