Inverse Functions

Inverse Functions Functions One-to-one Function Functions A function is a mapping whereby every element in the domain is mapped to only 1 element in the range. ie) Whatever number you start with, there is only 1 possible answer to the operation performed on it. An example of a mapping which is not a function would be square rooting, where the starting number may result in no answer, or 2 answers. Set A Set B eg) f(x) = x + 5 eg) f(x) = 3x - 2 Many-to-one Function Set A Set B eg) f(x) = x2 + 1 eg) f(x) = 6 - 3x2 Not a function Set A Set B eg) f(x) = √x eg) f(x) = 1/x 2B

‘A value in the domain (x) gets mapped to one value in the range’ Functions One-to-one Function Functions A function is a mapping whereby every element in the domain is mapped to only 1 element in the range. ie) Whatever number you start with, there is only 1 possible answer to the operation performed on it. An example of a mapping which is not a function would be square rooting, where the starting number may result in no answer, or 2 answers. ‘A value in the domain (x) gets mapped to one value in the range’ Many-to-one Function ‘Multiple values in the domain (x) get mapped to the same value in the range’ Not a function ‘A value in the range can be mapped to none, one or more values in the range’ 2B

Note: For a function to have an inverse, the members of its domain and range must be of one-to-one correspondence

Functions 2E Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) Some simple inverses Function Inverse f(x) = x + 4 f-1(x) = x - 4 g(x) = 2x g-1(x) = x/2 h(x) = 4x + 2 h-1(x) = x – 2/4 2E

Functions 2E Find the inverse of the following function Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) To calculate the inverse of a function, you need to make ‘x’ the subject f(x) = 3x2 - 4 y = 3x2 - 4 + 4 y + 4 = 3x2 ÷ 3 (y + 4)/3 = x2 Square root √((y + 4)/3) = x The inverse is written ‘in terms of x’ f-1(x) = √((x + 4)/3) 2E

Functions 2E Find the inverse of the following function Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) To calculate the inverse of a function, you need to make ‘x’ the subject m(x) = 3/(x – 1) y = 3/(x – 1) Multiply by (x – 1) y(x – 1) = 3 Multiply the bracket yx - y = 3 Add y yx = 3 + y Divide by y x = (3 + y)/y The inverse is written ‘in terms of x’ m-1(x) = (3 + x)/x 2E

The inverse is written ‘in terms of x’ Functions Finding f-1(x) Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) ‘The function f(x) is defined by f(x) = √(x – 2), x ε R, x ≥ 2’ ‘Find f-1(x), stating its domain. Sketch the graphs and describe the link between them. f(x) = √(x – 2) y = √(x – 2) Square y2 = x - 2 Add 2 y2 + 2 = x f-1(x) = x2 + 2 The inverse is written ‘in terms of x’ 2E

Finding the domain of f-1(x) Functions Finding the domain of f-1(x) Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) ‘The function f(x) is defined by f(x) = √(x – 2), x ε R, x ≥ 2’ ‘Find f-1(x), stating its domain. Sketch the graphs and describe the link between them. f-1(x) = x2 + 2 ‘The domain and range of a function switch around for its inverse’ f(x) f(x) = √(x – 2) x Range for f(x)  f(x) ≥ 0 or y ≥ 0 Domain for f-1(x)  x ≥ 0

Sketching the graph of f-1(x) Functions Sketching the graph of f-1(x) Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) ‘The function f(x) is defined by f(x) = √(x – 2), x ε R, x ≥ 2’ ‘Find f-1(x), stating its domain. Sketch the graphs and describe the link between them. f-1(x) = x2 + 2, {x ε R, x ≥ 0} Domain is x ≥ 0, so we can draw the graph for any values of x in this range f-1(x) = x2 + 2 f(x) f(x) = √(x – 2) x The link is that f(x) is reflected in the line y = x 2E

‘The domain and range of a function switch around for its inverse’ Inverse Functions You need to be able to work out the inverse of a given function. If f(x) is the function, the inverse is f-1(x) If g(x) is defined as: g(x) = 2x – 4, {x ε R, x ≥ 0}, Calculate and sketch g(x) and g-1(x), stating the domain of g-1(x). g(x) = 2x - 4 g-1(x) = (x + 4)/2 Domain  x ≥ 0 Domain  x ≥ -4 Range  g(x) ≥ -4 Range  g-1(x) ≥ 0 g(x) f(x) g-1(x) x The link is that f(x) is reflected in the line y = x

Pg 25 Exercise 4 Questions 1, 2, 8, 9 For homework look at slide 9,10,11 and make a final note on inverse functions in your jotter.