, g : x x 6, Sketch, in a single diagram, the graphs of y = f(x) and y = f -1 (x), making clear the

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Patricia Ellis
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1 PAST QUESTION ON FUNCTIONS 1. Express x + 4x in the form (x + a) + b, stating the numerical values of a and b. The functions f and g are defined as follows : f : x x 4x, x, g : x x 6, x R (ii) (iii) Show that the equation gf(x) = 0 has no real roots. State the domain of f -1, and find an expression in terms of x for f -1 (x). Sketch, in a single diagram, the graphs of y = f(x) and y = f -1 (x), making clear relationship between these graphs. Specimen paper 001. The function f : x x 5x 8 is defined for the domain x a, where a is a constant. Express x 5x 8 in the form ( x p) q. (ii) Find the smallest value of a for which f has an inverse. (iii) Find the domain of f -1 corresponding to this value of a. June Express 1x 11 x in the form a x b c. [3] (ii) Given that f : x x 1x 11, for the domain x 0, find the range of f. [] Nov 001 Q 4. The functions f and g are related by f : x 3x, x, 6 g : x, x 3 x, x 1. 5 Find the value of x for which fg(x) = 3. [3] (ii) Sketch, in a single diagram, the graphs of y = f(x) and y = f -1 (x), making clear the relationship between the two graphs. [3] (ii) Express each of f -1 (x) and g -1 (x) in terms of x, and solve the equation f -1 (x) = g -1 (x). [5] June 00 Q10 5. Express x + 8x 10 in the form a(x + b) + c. [3] (ii) For the curve y = x + 8x 10, state the least value of y and the corresponding value of x. [] (iii) Find the set of values of x for which y 14. [3] Given that f : x x 8x 10 for the domain x k, (iv) find the least value of k for which f is one-one, [1] (v) express f -1 (x) in terms of x in this case. [3] Nov 00 Q11 MES LAILA PTEK 011 1

2 6. The function f is defined by f : x _ ax + b, for x, where a and b are constants. It is given that f() = 1 and f(5) = 7. Find the values of a and b. [] (ii) Solve the equation ff(x) = 0. [3] June 003 Q5 7. The equation of a curve is y = 8x x. Express 8x x in the form a (x + b), stating the numerical values of a and b. [3] (ii) Hence, or otherwise, find the coordinates of the stationary point of the curve. [] (iii) Find the set of values of x for which y 0. [3] The function g is defined by g : x 8x x, for 4 x. (iv) State the domain and range of g -1. [] (v) Find an expression, in terms of x, for g -1 (x). [3] June 003 Q11 8. Functions f and g are defined by f : x x 5, x, 4 g : x, x x, x. Find the value of x for which fg(x) = 7. [3] (ii) Express each of f -1 (x) and g -1 (x) in terms of x. [3] (iii) Show that the equation f -1 (x) = g -1 (x) has no real roots. [3] (iv) Sketch, on a single diagram the graphs of y = f(x) and y = f -1 (x), making clear the relationship between these two graphs. [3] Nov 003 Q10 9. The functions f and g are defined as follows : f : x x x, x, g : x x 3, x. Find the set of values of x for which f(x) 15. [3] (ii) Find the range of f and state, with a reason, whether f has an inverse. [4] (iii) Show that the equation gf(x) = 0 has no real solutions. [3] (iv) Sketch, in a single diagram, the graphs of y = g(x) and y = g -1 (x), making clear the relationship between the graphs. [] June 004 Q10 MES LAILA PTEK 011

3 10. The function f : x x a, where a is a constant, is defined for all real x. In the case where a = 3, solve the equation ff(x) = 11. [3] The function g : x x 6x is defined for all real x. (ii) Find the value of a for which the equation f(x) = g(x) has exactly one real solution. [3] The function h : x x 6x is defined for the domain x 3. (iii) Express x 6x in the form (x p) q, where p and q are constants. [] (iv) Find an expression for h -1 (x) and state the domain h -1. [4] Nov 004 Q9 11. A function f is defined by f : x (x 3) 3 8, for x 4. (ii) Find an expression, in terms of x, for f 1 (x) and find the domain of f 1. [4] 1. Functions f and g are defined by Q8 (ii)nov 005 Find the values of k for which the equation f(x) = g(x) has two equal roots and solve the equation f(x) = g(x) in these cases. [6] (ii) Solve the equation fg(x) = 5 when k = 6. [3] (iii) Express g -1 (x) in terms of x. [] June 006 Q The function f is defined by. Find the set of values of x for which f(x) > 4. [3] (ii) Express f(x) in the form, stating the values of a and b [] (iii) Write down the range of f. [1] (iv) State, with a reason, whether f has an inverse. [1] The function g is defined by (v) Solve the equation g(x) = 10. [3] Nov 006 Q10 MES LAILA PTEK 011 3

4 14. The diagram shows the graph of y = f(x), where f : x for x 0. (ii) Find an expression, in terms of x, for f -1 (x) and find the domain of f -1. [4] (iii) Copy the diagram and, on your copy, sketch the graph of y = f -1 (x), making clear the relationship between the graphs. [] The function g is defined by g : x ½ x for x 0. (iv) Solve the equation fg(x) =. [3] 15. The function f is defined by f : x _ x 8x + 11 for x. Jun 007 Q11 Express f(x) in the form a(x + b) + c, where a, b and c are constants. [3] (ii) State the range of f. [1] (iii) Explain why f does not have an inverse. [1] The function g is defined by g : x _ x 8x + 11 for x A, where A is a constant. (iv) State the largest value of A for which g has an inverse. [1] (v) When A has this value, obtain an expression, in terms of x, for g -1 (x) and state the range of g -1. [4] Nov 007 Q Functions f and g are defined by f : x 4x k for x, where k is a constant, g : x for x, x. Find the values of k for which the equation fg(x) = x has two equal roots. [4] (ii) Determine the roots of the equation fg(x) = x for the values of k found in part. [3] Jun 008 Q8 MES LAILA PTEK 011 4

5 17. The function f is defined by f : x 3x for x. Sketch, in a single diagram, the graphs of y = f(x) and y = f -1 (x), making clear the relationship between the two graphs. [] The function g is defined by g : x _ 6x x for x. (ii) Express gf(x) in terms of x, and hence show that the maximum value of gf(x) is 9. [5] The function h is defined by h : x _ 6x x for x 3. (iii) Express 6x x in the form a (x b), where a and b are positive constants. [] (iv) Express h -1 (x) in terms of x. [3] Nov 008 Q The function f is defined by f : x. x 1x + 13 for 0 x A, where A is a constant. Express f(x) in the form a(x + b) + c, where a, b and c are constants. [3] (ii) State the value of A for which the graph of y = f(x) has a line of symmetry. [1] (iii) When A has this value, find the range of f. [] The function g is defined by g : x. x 1x + 13 for x 4. (iv) Explain why g has an inverse. [1] (v) Obtain an expression, in terms of x, for g -1 (x). [3] 19. The function f is such that f(x) = for x >, x.5. Jun 009 Q10 (ii) Obtain an expression for f -1 (x). [] 0. The function f : x. x 8x + 14 is defined for x. Nov 009 Q8 Find the values of the constant k for which the line y + kx = 1 is a tangent to the curve y = f(x). [4] (ii) Express f(x) in the form a(x + b) + c, where a, b and c are constants. [3] (iii) Find the range of f. [1] The function g : x. x 8x + 14 is defined for x A. (iv) Find the smallest value of A for which g has an inverse. [1] (v) For this value of A, find an expression for g 1 (x) in terms of x. [3] MES LAILA PTEK 011 5

6 Jun 010 Q10 1. The diagram shows the function f defined for 0 x 6 by for 0 x, for < x 6. State the range of f. [1] (ii) Copy the diagram and on your copy sketch the graph of y = f 1 (x). [] (iii) Obtain expressions to define f 1 (x), giving the set of values of x for which each expression is valid. [4] Nov 010 Q7 MES LAILA PTEK 011 6

Math 110 Midterm 1 Study Guide October 14, 2013

Math 110 Midterm 1 Study Guide October 14, 2013 Name: For more practice exercises, do the study set problems in sections: 3.4 3.7, 4.1, and 4.2. 1. Find the domain of f, and express the solution in interval notation. (a) f(x) = x 6 D = (, ) or D = R