Skill Sets Chapter 5 Functions No. Skills Examples o questions involving the skills. Sketch the graph o the (Lecture Notes Example (b)) unction according to the g : x x x, domain. x, x - Students tend to sketch the graph without reerring to the domain given by the unction.. Find the range o the unction by drawing good graphs. - Students must realize that the accuracy o the range ound depends on accurate graphs. Do pay attention to any asymptotes, turning points other eatures o the graph. 3. Determine i the unction is one to one. (Lecture Notes Example (c)) x h : x e, x, x 0 R h = (0,] Here the horizontal asymptote is an important eature or iguring out the range. When the unction is one to one: (Lecture Notes Example 3(b)) g : x cos x, x, 0 x, Method : By applying the Horizontal Line Test stating the correct reason to justiy whether the unction is one to one.
Every horizontal line intersects the graph o x y g at no more than one point, thus the unction g is a one to one unction. Thereore its inverse exists. When the unction is not one to one: (Lecture Notes Example 3(c)) h : x x, x. Draw a horizontal line on your graph to show that it intersects the curve at two points. Thus h is not a one to one unction. 4. Find the rule o the inverse unction by letting y = (x) making x as the subject eventually. (a) When (x) is a quadratic unction. OR Give a counterexample like x, x 3 gives the same y value = 0. (Lecture Notes Example 5) : x x x 3, or x. Method : Complete square From y x x, we have y 4 y x x 3 x y 4. Since x, thereore we take x y 4. Thus y y x 4 x 4. Method : Express the quadratic expression in the orm o From y x, we have y x x 3
ax bx c 0 ind x in terms o y using b b 4ac x. a x x y 3 0 4 4()( 3 y) x 4(4 y) x x 4 y Since x, thereore we take x y 4. Thus y y x 4 x 4. (b) When (x) is a logarithmic unction - Realize that inverse o a ln unction is the exponential unction. (Tutorial Q(d)) : x ln x, x, x 0 Let y lnx y x x x x e y ln x ln e [Note: e ] y x ( x) e. (c) When (x) is a modulus unction - Resolve modulus unction into expressions. i.e. x ( x ) or ( x ) 5. Find the domain range o the inverse unction using the relation: (Tutorial Q(e)) : x, x x, x Let y x Then, y since x. x x y x x y y ( x). x Domain o Range o Range o Domain o 3
6. Sketch the graphs o y = (x) y = - (x) on the same diagram Important to note: (a) change o coordinates rom (x, y) to (y, x) ater relecting the graph o y = (x) in the line y = x, (Lecture Notes Example 4) (b) same scale on both axes. 7. Restrict the domain o so that - exists. (Lecture Notes Example 5(i)) : x x x 3, or x From the graph, we observe that or the inverse to exist, i.e. or is one to one, the largest domain we can go or is x. Thus the smallest value o p is. 8. State the range o domain o g to check the existence o composite unction g. 9. Find the rule o composite unction g its domain Students need to check R Dg or g exists. Students must state the range o domain o g explicitly beore concluding i the ormer is a subset o the latter. (Extension o lecture notes example 6) : x x, x, x 0 g : x x, x g ( x) g( ( x)) g( x) ( x) x Dg D [0, ) 4
0. Find the range o g using Method : Do it stages. g D D R R g g (Lecture Notes Example 6) : x x, x, x 0 g : x x, x D D R g g [0, ) [, )? Use the graph o y = g(x), substitute [, ) into the unction g. y y = g(x) x Rg (,0] Method : Graphical method - The graph o g may not be easy to draw accurately. - This method is recommended i g is a simple unction. (Lecture Notes Example 7(iii)) Sketch the graph o y = g(x) according to the domain o g. Find the range o g. y g x, x. x e Sketch the graph o the composite unction g subjected to its domain. From the graph, we obtain the range o g,. 5
. Restrict the domain o g so that Rg Dor composite unction g to exist. (Lecture Notes Example 8) g : x ln x, x, x, x h : x e, x. - Basically we crop the unction o h so that the new R h D g. Rh (, ), Dg (, ) Since Rh Dg, gh does not exist. For gh to exist, Rh (, ). The greatest possible range o h,. From the graph, the greatest possible domain o h or gh to exist is (, 0).. Solving the equation (x) = - (x) is the same as solving (x) = x. (Tutorial Q(iii)) Since the graphs o y = (x), y = - (x) y = x will meet at the same point, to ind the exact solution o the equation (x) = - (x), we could solve (x) = x. i.e., 3x x. 3. Know the dierence between the graph o (Tutorial Q6(iii)) Given :, x x xir, x What is the dierence between the graph o? ( ) ( ) x x x [This result is true all the time, regardless o ]. Thereore rule. However, the domain o domain o = domain o 6 share the same = domain o = (,) while (, ). You should also realize that are composite unctions too they always exist. Check the condition or
the existence o composite unction yoursel. 4. You should also use the summary on Page 0 to help you revise through the topic. 5. The learning objectives at the irst page o the lecture notes is also good or knowing the important concepts that are taught in this chapter. 7