RVHS H2 Math Functions Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Functions 1 Functions 1. AJC/I/5/2009 The function f is defined by 2 2f : e 2 xxx −→− , x < 0. (i) Find -1f in a similar form. [3] The function g is defined by 3g: 1 xx x −→ − , xk , 1x . (ii) Find the least value of k such that the function -1fg exists. [2] (iii) Hence find the range of -1fg , giving your answer in exact form. [1] 2. IJC/I/10 (a) The function f is defined by ( )1f : 2 tan 2x x x −−  , 33 22 x−   . (i) Show that f(x) decreases as x increases. [3] (ii) Find the range of f. [1] (iii) Explain whether the composite function ff exists. [2] (iv) The curve C is obtained by reflecting the graph of f in the line yx= . Write down the equation of C. [1] (b) The function h is defined by 2h : 2xx +  , 𝑥 ∈ ℝ. The function g is such that the composite function gh exists and ( ) 42gh 2 5x x x= − + , 𝑥 ∈ ℝ. Find an expression for ( )g x . [2] 3. NYJC/I/2 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 and define fg in similar form. [2] (ii) If h is a function defined such that h( x) = gfg( x) for all 0x  , show that the composite function gfg exists. Hence, find an expression for h(x). [2] (iii) Solve ( ) ( )-1h g gxx= , giving your answer in exact form. [2]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Functions 2 4. PJC/I/11 Three functions f, g, h are defined as follows: f : 2 xx  , 𝑥 ∈ ℝ 1g : 2 xx −  , 𝑥 ∈ ℝ 2h : 2x x x +−  , 𝑥 ∈ ℝ (i) Describe a single geometrical transformation which maps the graph of f onto the graph of g. [1] (ii) Sketch the graph of h and explain why 1h− does not exist. [2] (iii) Write down the value of a such that {𝑥 ∈ ℝ,  𝑥 > 𝑎}is the largest possible domain of h for which 1h− exists, and hence define 1h− in a similar form. [4] (iv) Using the domain of h found in (iii), explain why 1hf− exists. Define 1hf− in a similar form and find its range. [4] 5. YJC/1/8 The functions f and g are defined as follows : f : x → )2ln( +x , 2−x , g : x → )(f x , 𝑥 ∈ ℝ . (a) Solve for all the values of x satisfying the equation fg(x) = gf(x). [4] (b) (i) Briefly explain why the inverse function of g cannot be formed. [1] (ii) The function g has an inverse if its domain is restricted to ax . Determine the largest value of a. [1] (iii) Define completely the inverse function 1g− corresponding to the largest value of a determined in part (b)(ii). [3] 6. MJC/2013/I/6 The functions f and g are defined as follows: f: 𝑥 ↦ 𝜆 − (𝑥 + 2)2, 𝑥 ∈ ℝ, 𝑥 > −2, g: 𝑥 ↦ ln(1 − 𝑥) , 𝑥 ∈ ℝ, 𝑥 < 1, where  is a constant. (i) Given that gf exists, find the largest value of  . [2] In the rest of the question, use the value of  found in part (i). (ii) On the same diagram, sketch the graphs of ( ) ( ) 1f , fy x y x −== and ( ) 1ffyx −= , showing clearly the relationship between the graphs. [3] (iii) Hence find the exact solution of ( ) ( ) 1ff xx −= . [3]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Functions 3 7. ACJC/2011/I/9 Sketch the curve given by the equation 22 5y ax+= for 0x and 0y , where a is a positive constant. [1] The functions f and g are defined by 2f : 5x ax −  , 𝑥 ∈ ℝ,   0 ≤ 𝑥 ≤ √ 5 𝑎 g: 𝑥 ↦ 1 + 𝑒−𝑥, 𝑥 ∈ ℝ,  𝑥 ≥ 0. Show that 1f − exists and define 1f − in a similar form. [4] Given that f 2(𝑥) = 𝑥, for all 𝑥 ∈ ℝ,   0 ≤ 𝑥 ≤ √ 5 𝑎 , (i) show that a = 1 without evaluating 2f ( )x , [2] (ii) using the result in (i), show that fg exists and find its corresponding range. [3] (iii) Sketch the curve given by the equation 22 5y ax+= for 0x and 0y , where a is a positive constant. [1] 8. DHS Prelim/2020/02/Q1 (a) The functions f and g are defined as follows f : 7 ,xx −  where 𝑥 ∈ ℝ, 𝑥 < 7, ( ) 4 2g : 2 2 ,x x x −+  where 𝑥 ∈ ℝ, 𝑥 > −2. (i) Show that the composite function gf exists and define gf in a similar form. [2] (ii) Find the range of gf. [2] (b) The functions k and kh are defined by k : 5,xx −  where 𝑥 ∈ ℝ, 2kh : ,x x a +  where 𝑥 ∈ ℝ, 𝑥 > √5, where a is a positive constant. Leave your answers for the following parts, (b)(i) and (ii), in terms of a. (i) Find h(x). [2] (ii) Explain why 1h− exists. Hence or otherwise, find 1h ( )x− and state the domain of 1h.− [3] 9. NYJC/2014/I/1 The function f is defined by f: 𝑥 ↦ 2 − (𝑥 + 1)2, 𝑥 ∈ ℝ, 𝑥 ≤ 𝑘. (i) Determine the largest value of k for which the function 1f − exists. [1] With this value of k, (ii) find 1f − in a similar form. [3] (iii) show algebraically that the x-coordinate of the point of intersection of the curves ( ) ( )1f and fy x y x −== satisfies the equation 2 3 1 0xx+ − = and find the value of this x-coordinate, correct to 3 decimal places. [2]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Functions 4 10. IJC/2013/I/13 It is given that 2 3 for 2 1,(2 3) 2f ( ) 3 for 1 0, xxx x  −  − +−=  − −   and that f ( ) f ( 2)xx=+ for all real values of x. (i) Sketch the graph of f ( )yx= for 55 22 x−   . [3] (ii) Find the exact value of 1 3 2 f ( ) dxx − . [4] The function h is defined by 2 3h : for 2(2 3) 2x x a x −  +−  . (iii) Write down the greatest value of a such that 1h− exists. [1] Assume that a takes the value found in part (iii). (iv) Sketch, on a single diagram, the graphs of h( )yx= and 1h ( )yx −= . [2] (v) Explain why the x-coordinate of the point of intersection of the curves in part (iv) satisfies the equation 324 12 7 3 0x x x+ + − = , and find the value of this x-coordinate, correct to 4 significant figures. [3] 11. AJC/2018/I/9 The functions f and g are defined by f: 𝑥 ↦ (𝑥 + 3)|𝑥 − 2| , 𝑥 ∈ ℝ, g: 𝑥 ↦ 24 𝑥2 + 3 − 1 , 𝑥 ∈ ℝ, 𝑥 > 0. (i) Explain why the function 1f − does not exist. [1] In the rest of the question, the domain of f is restricted to [k, 2] where k is the least value such that 1f− exists. (ii) Write down the value of k. Find 11f ( ) and state the domain of f .x−− [4] (iii) Sketch the graphs of )(f xy = and 1f ( )yx −= on the same diagram, showing clearly the relationship between the two graphs. Hence, find the exact solution of the equation 1f ( ) f ( )xx −= . [4] (iv) Determine whether 1g f exists.− [2] Show that g is a strictly decreasing function. Hence, without finding 1g− , find the range of values of x for which 1g (f ( )) 1,x−  giving your answer in exact form. [3]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Functions 5 12. ACJC Prelim/2020/01/Q6 The function f is defined by ( ) 2 1 3 1 for 0 2, : 2 4 for 2 4. f xx x xx  − −   − −    It is given that f ( 4) f ( )xx+= for all real values of x. (i) Sketch the graph of f ( )yx= for 19 x−   , stating the coordinates of the end points. Find the range of f. [4] The function g is defined by