Inverse of a function
As we approach exams, the importance of systematic practice cannot be overstated. Please be sure that you are involved in this vital exercise.
We began the review of functions and relations in the last lesson when the fundamentals of the topic were introduced. In this week’s lesson, we will initially share the solution to the homework given. Given that f : x ----> 7x - 1 g : x ----> 2x + 9 Evaluate: (i) g(- 3) (ii) fg(2) Solution: (i) Since g : x ----> 2x + 9 then g(x) = 2x + 9 g(-3) =(2 x -3) + 9 =-6 + 9 = 3. g(-3) = 3 As g(x) = 2x + 9 and f(x) = 7x - 1 fg(x) = f (2x + 9) g(x) or (2x + 9) replaces x in f(x) fg(x) = 7 (2x +9) - 1
= 14x + 63 - 1 = 14x + 62 fg(2) = (14 x2) + 62 = 28 + 62 = 90 fg(3) = 90 If f(x) = 2x + 5 and g(x) = 1/2 (x - 3), Calculate : (i) f(19) (ii) gf(3) Solution: (i) As f(x) = 2x + 5, then f(19) =(2 x 19) + 5 = 38 + 5 = 43. f(19) = 43 (ii) f(3) =(2 x 3) + 5 = 11, Since f(3) = 11 then gf(3) = g(11) f(3) or 11 replaces x in g(x) Since g(x) = 1/2 (x - 3) g(11) = 1/2 (11 - 3 ) = 8/2 = 4 gf(3) = 4 . Alternatively f(x) = 2x + 5 and g(x) = 1/2 (x - 3), gf(x) = g (2x + 5) = 1/2 ((2x +5) - 3) = 1/2 (2x + 5 - 3) gf(x) = 1/2 (2x + 2) gf(3) = 1/2 (2x 3 + 2) gf(3) = 8/2 = 4 Now that we have gone through the homework, our review of the topic will continue.
INVERSE OF A FUNCTION
Specific Objective Given the function f, find its inverse f- 1 If f is the function defined as y = ax + b, then f- 1 is the inverse function and it expresses the variable x in terms of y. Example: y = ax + b ax =y - b x= y - b ( x is expressed as a function of y) a Interchange x for y. (This is necessary as y is always expressed as a function of x.) y=x-b a f- 1( x)= x - b or f- 1 =x - b a a that is, f- 1, the inverse of function f, is x- b a Please note that this method should always end with the statement: f- 1 (x)= x - b and never y= x - b a a Given the function y = ax + b, some students express f- 1 (x) as 1 by assuming that -1 is the power of f as in indices. ax + b I am sure you will never make this error. Example 1: Given that f(x) = 1/2(2x + 3). Calculate f- 1 (x) Solution Since f(x) = 1/2(2x + 3) y = 1/2(2x + 3) 2y = 2x + 3 2x = 2y - 3 x = 1/2(2y - 3) Interchanging x for y (Always remember this step. It must also be explicitly stated.) y = 1/2(2x - 3) f- 1( x) = 1/2(2x - 3) Example 2: Given f(x) = 1/2x and g(x) = 3x - 2 Calculate: (i) g(-2) (ii) fg (4) (iii) f- 1( 4) Solution (i) Since g(x) = 3x - 2, then g(-2) =3 x -2 -2 = - 6 - 2 = - 8 Note x is replaced by -2 in g(x). g(-2) = - 8 (ii) Given g(x) = 3x - 2 then fg (x) = f(3x - 2) fg (4) = f(3x4 - 2) = f(10) As f(x) = 1/2x f(10) = 10/2 = 5. fg (4) = 5 I am sure that you can now show that fg(x) = 3x - 2 2 (iii) As f(x) = 1/2x then y = x/2 x = 2y. Interchanging x for y y = 2x f- 1( x) = 2x f- 1( 4) = 8 Please be sure that you are comfortable with the methods of cross-multiplication and changing the subject of a formula.
INVERSE OF A COMPOSITE FUNCTION
Given the functions y = f(x) and y = g(x), then y = gf(x) is a composite function. Since gf(x) is a function of x, the inverse is found by using the method outlined above. Example: Given the functions f(x) = 3x and g(x) =x - 2, determine the functions: (a) fg(x) (b) [fg]- 1( x)
Solution: (a) As f(x) = 3x and g(x) = x - 2 fg(x) = f(x - 2) = 3(x - 2) fg(x) = 3(x - 2) (b)y = fg(x) = 3(x - 2) y = 3(x - 2) = 3x - 6 3x =y + 6 x= y + 6 Interchange x for y 3 y=x+6 3 The inverse of fg(x) OR fg -1(x) is x + 6 3 Let us attempt another example: Example Given f(x) = x2 and g(x) = 5x + 3, calculate (i) f(-2) (ii) (g f) -1x Solution (i) Since f(x) = x2 f(-2) = (-2)2 = 4 Answer: f(-2) = 4 (ii) Given that f(x) = x2 and g(x) = 5x + 3 then gf(x) = g(x2) Since g(x) = 5x + 3 g(x2) = 5x2 + 3 gf(x) = 5x2 + 3
HOMEWORK
f and g are functions defined as follows f : x ----> 4x + 3
2 g: x ----> 3x + 5 (a) Calculate the value of f (-3) (b) Write expressions for (i) f -1(x) (ii) g-1 (x) (c) Hence, or otherwise, write an expression for (gf) -1 Given that f : x 5x + 3 and g : x 2x + 1 (a) Determine fg-1(x) and g-1 f-1 (x) (b) Hence evaluate ( fg )-1 (5 ) and g-1 f-1 (5) . Have a good week