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- 1. FUNCTIONS 1 Function Basics 1.1 Set notations [a, b] = {x ∈ R : a ≤ x ≤ b} (a, b] = {x ∈ R : a < x ≤ b} (a, b) = {x ∈ R : a < x < b} (a, ∞) = {x ∈ R : x > a} R = set of real numbers R+ = set of positive real numbers 1.2 Rule, domain and range A function f, is defined by its rule and domain. f : x → x + 2 rule , x ∈ [0, ∞). domain f(x) = x + 2, x ≥ 0. X f(X) 1 2 3 3 4 5 f Domain Range The domain Df , is the set of all possible x values. The range Rf , is the set of all possible y values. Remark: When we state a function, we must always state both its rule and domain. www.MathAcademy.sg 1 © 2019 Math Academy
- 2. Finding Range Sketch the graph to find Rf . Rf is the range of y-values that the graph takes. Example 1. Find the range of the following: (a) f : x → x2 − 1, x ∈ [−1, 2], −1 1 2 −1 3 x y ∴ Rf = (b) g : x → ex + 1, x ∈ R. 1 x y ∴ Rg = 2 Horizontal line test A function is said to be 1 − 1 if for every y ∈ Rf , there is only ONE x such that f(x) = y. −1 1 1 x y It is not 1-1 since the line y = 1 cuts the graph twice. x y It is 1-1 since every horizontal line cuts the graph at most once. Horizontal Line Test f is NOT a 1 − 1 function if there is a horizontal line that cuts the graph at MORE THAN ONE point. f IS a 1 − 1 function if any horizontal line y = a, where a ∈ Rf , cuts the graph AT ONLY ONE point. Differentiation test for 1-1 f is 1 − 1 if f′(x) > 0 for ALL x in the domain (strictly increasing functions) or f′(x) < 0 for ALL x in the domain (strictly decreasing functions) www.MathAcademy.sg 2 © 2019 Math Academy
- 3. Example 2 (Different techniques to answer 1-1 questions). [2012/RVHS/I/6modified] The function f is defined by f : x → | − x2 − 2x + 3|, x ∈ R. (a) With the aid of a diagram, explain why f is not 1-1. [2] (b) If the domain of f is restricted to the set {x ∈ R : x ≥ k}, state with a reason the least value of k for which the function is 1-1. [2] (c) By considering the derivative of f(x), prove that f is a one-one function for the domain you have found in (b). [2] Solution: (a) Show not 1-1 through graph Sketch the graph, give the equation of a SPECIFIC horizontal line that cuts the graph in at least 2 points. −3 1 3 y = f(x) x y Solution: (b) Show 1-1 through graph Sketch the graph, explain that any horizontal line cuts the graph at only 1 point. Least value of k is 1. When x ≥ 1, any horizontal line y = a for a ∈ [0, ∞), cuts the graph of y = f(x) at only 1 point, hence it is 1-1. (c) Show 1-1 through differentiation Show that f′(x) is either > 0 or < 0 for all x ∈ Df . Note: This method cannot be used to show that a function is not 1-1. For x ≥ 1, f(x) = x2 + 2x − 3. Since f′(x) > 0 for x ≥ 1, f is a strictly increasing function, and hence it is 1-1. www.MathAcademy.sg 3 © 2019 Math Academy
- 4. 3 Inverse functions Df Rf x y f f(x) = y f−1 f−1(y) = x Properties of inverse function 1. For f−1 to exist, f must be a 1-1 function. 2. Df−1 = Rf . 3. Rf−1 = Df . 4. (f−1)−1 = f. Geometrical relationship between a function and its inverse (i) The graph of f−1 is the reflection of the graph f about the line y = x. (ii) (a, b) lies on f ⇔ (b, a) lies on f−1. (iii) x = k is an asymptote of f ⇔ y = k is an asymptote of f−1 Remark: The notation f−1 stands for the inverse function of f. It is not the same as 1 f . www.MathAcademy.sg 4 © 2019 Math Academy
- 5. Example 3 (Cambridge N2008/II/4). The function f is defined by f : x → x2 − 8x + 17 for x > 4. (i) Sketch the graph of y = f(x). Your sketch should indicate the position of the graph in relation to the origin. (ii) Show that the inverse function f−1 exists and find f−1(x) in similar form. (iii) On the same diagram as in part (i), sketch the graph of y = f−1. (iv) Write down the equation of the line in which the graph of y = f(x) must be reflected in order to obtain the graph of y = f−1, and hence find the exact solution of the equation f(x) = f−1(x). Solution: (i) 4 1 y = f(x) x y (ii) Showing inverse exists To show f−1 exists, we only need to show that f is 1-1. Every horizontal line y = a for a > 1 cuts the graph of y = f(x) at only 1 point, hence it is 1-1 and f−1 exists. [We first make x the subject:] ∴ f−1(x) = √ x − 1 + 4, x > 1. [Change x to f−1x and y to x.] [Note: You must state the domain of f−1 !] www.MathAcademy.sg 5 © 2019 Math Academy
- 6. (iii) Geometrical relationship between a function and its inverse (i) The graph of f−1 is the reflection of the graph f about the line y = x. (ii) (a, b) lies on f ⇔ (b, a) lies on f−1. (iii) x = k is an asymptote of f ⇔ y = k is an asymptote of f−1 1 4 1 4 y = f(x) y = f−1(x) y = x x y (iv) It must be reflected along the line y = x. Since f and f−1 intersect at the line y = x, finding exact solution of the equation f(x) = f−1(x) is equivalent to finding f(x) = x x2 − 8x + 17 = x x2 − 9x + 17 = 0 x = 9 ± √ 92 − 4(1)(17) 2 = 9 + √ 13 2 or 9 − √ 13 2 (rej as it is not in Df ) www.MathAcademy.sg 6 © 2019 Math Academy
- 7. Example 4 (2010/MI/Prelim/I/2c). Function g is defined by g : x → x2 − 3x for x ∈ R, If the domain of g is restricted to the set {x ∈ R : x ≥ a}, find the least value of a for which g−1 exists. Hence, find g−1 and state its domain. [4] [a = 1.5, g−1(x) = 3 2 + √ x + 9 4, x ≥ −9 4] www.MathAcademy.sg 7 © 2019 Math Academy
- 8. Example 5 (2014/PJC/Prelim/I/7modified). It is given that f(x) = { 1 1+ √ x for 0 ≤ x < 4, −1 for 4 ≤ x < 5, and that f(x + 5) = f(x) for all real values of x. (i) Find f(24) and f(30). (ii) Sketch the graph of y = f(x) for −5 ≤ x < 12. Solution: (i) (ii) −5 5 10 12 1 -1 1 3 y x 4 www.MathAcademy.sg 8 © 2019 Math Academy
- 9. 4 Composite Functions 4.1 Domain, Range, Rule Let f and g be the following functions: g : x → x + 1 for x ≥ −1, f : x → x − 2 for x ≥ 0. Lets investigate the composite function fg(x). -1 0 1 2 0 1 2 3 ... ... g Dg Rg What happens now if we change the domain of f to the following? g : x → x + 1 for x ≥ −1, f : x → x − 2 for x ≥ 1. -1 0 1 2 0 1 2 3 ... ... g Dg Rg At the intermediate stage, the number 0 will have no place to map to! This is an undesired situation. www.MathAcademy.sg 9 © 2019 Math Academy
- 10. fg Diagram for function fg(x) Dg Rfg Df Rg Properties of Composite Functions (i) Domain fg = domain g. (ii) Composite function fg exists ⇔ Rg ⊆ Df Example 6. Consider the following functions: f : x → x2 x ∈ R g : x → 1 x − 3 x ∈ R, x ̸= 3 Find the composition functions (i)fg, (ii) gf. Solution: (i) fg(x) = f (g(x)) = f ( 1 x − 3 ) = ( 1 x − 3 )2 Domain of fg(x) = domain g = R {3}. (ii) gf(x) = g (f(x)) = g(x2 ) = 1 x2 − 3 Domain of gf(x) = domain f = R. www.MathAcademy.sg 10 © 2019 Math Academy
- 11. Example 7. Two functions, f, g are defined by f : x → x2 , x ≥ 0 g : x → 2x + 1, 0 ≤ x ≤ 1 (a) Show that the composite function fg exist. Find (b) its rule and domain and (c) corresponding range. Solution: (a) To show that fg exists, we need to show that Rg ⊆ Df . 1 1 3 y = g(x) x y Since Dg = [0, 1], from the graph of g, we observe that Rg = [1, 3]. Rg = [1, 3] Df = [0, ∞) ⇒ Rg ⊆ Df Therefore, fg exists. (b) f(g(x)) = f(2x + 1) = (2x + 1)2 Dfg = Dg = [0, 1] (c) To find Rfg, Step 1. Draw the graph of f. Step 2. For the domain, set it as Range g, which in this case is [1, 3] Step 3. Find the range under this new domain. −1 1 3 1 9 y = f(x) x y Therefore, Rfg = [1, 9]. www.MathAcademy.sg 11 © 2019 Math Academy
- 12. Example 8 (2009/NYJC/Prelim/I/2a modified). The functions f and g are defined by f : x → x2 − 1, x ∈ R, g : x → √ x + 4, x ∈ R+ . (i) Show that the composite function fg exists. (ii) Define fg in similar form. (iii) Find the range of fg. [ii) fg(x) = x + 3, x ∈ R+ iii) (3,∞)] www.MathAcademy.sg 12 © 2019 Math Academy
- 13. 4.2 ff−1 (x) and f−1 f(x) A function given by f(x) = x is called an identity function. It maps any value to itself. f−1f Diagram for function f−1f(x) a b a x ff−1(x) = f−1f(x) = x Dff−1 = Df−1 . Df−1f = Df . Key Observations (a) Both ff−1(x) and f−1f(x) are equal to x. (b) ff−1(x) and f−1f(x) have different domains. (Sketch the correct domain on a graph) Example 9. Let the function f be defined by f : x → x + 4, x ∈ [0, ∞). Sketch the graphs of the functions ff−1 and, f−1f on 2 separate graphs. www.MathAcademy.sg 13 © 2019 Math Academy
- 14. Example 10. Consider the following functions: g : x → 1 x − 3 x ∈ R, x ̸= 3 g−1 h : x → 1 x , x ≥ 0 Find the function h. Solution: h = gg−1 h = g(g−1 h) = g ( 1 x ) = 1 1 x − 3 = x 1 − 3x Domain h = domain g−1 h = [0, ∞) Example 11. Consider the following functions: fg : x → x2 − 6x + 14 x ∈ R+ g−1 : x → x + 5, x ∈ R Find the function f. [f(x) = x2 + 4x + 9, x ∈ R] www.MathAcademy.sg 14 © 2019 Math Academy
- 15. Example 12 (Self-inverse Functions). The functions f and g are defined as follows: f : x → 5 − x 1 − x , x ∈ R, x ̸= 1. (i) Explain why f has an inverse, f−1, and show that f−1 = f. (ii) Evaluate f51(4). Solution: (i) f(x) = 5−x 1−x = 1 + 4 1−x . f′ (x) = 4(−1)(1 − x)−2 (−1) = 4 (1 − x)2 Since f′(x) > 0 for all x ∈ R, x ̸= 1, f is 1-1 and hence has an inverse. y = 1 + 4 1 − x y − 1 = 4 1 − x 1 y − 1 = 1 − x 4 4 y − 1 = 1 − x 1 + 4 1 − y = x ∴ f−1 (x) = 1 + 4 1 − x 1 1 x y From the graph of f, Rf = R{1}. ∴ Df−1 = R{1} = Df . Hence f = f−1. (ii) f(x) = f−1 (x) f2 (x) = x f3 (x) = f ( f2 (x) ) = f(x) f4 (x) = f ( f3 (x) ) = f(f(x)) = f2 (x) = x ... ∴ f51 (x) = f(x) ∴ f51 (4) = f(4) = 5 − 4 1 − 4 = −1 3 www.MathAcademy.sg 15 © 2019 Math Academy