Inverse Functions solutions February 11, 2019 Q:\1. Curriculum Subjects\Mathematics\Graphing Package AutoSave 1 Inverse Functions solutions February 11, 2019 Fact: If the function has an inverse function then this inverse function: • is denoted Note: is not the reciprocal of AutoSave i.e. 2 Inverse Functions solutions February 11, 2019 Fact: If the function has an inverse function then this inverse function: • is the reflection of Consider in the line f(x) y=x (Note: broken line) f 1( x ) Reflected point AutoSave 3 Inverse Functions solutions February 11, 2019 Fact: If the function • must be a function has an inverse function then this inverse function: i.e. must satisfy the vertical line test Consider vertical line test f(x) f 1( x ) AutoSave 4 Inverse Functions solutions February 11, 2019 Consider vertical line test f ( x ) is a function but f 1( x ) is not a function Thus, no inverse function AutoSave 5 Inverse Functions solutions February 11, 2019 Consider vertical line test Both f ( x ) and f 1( x ) are functions AutoSave 6 Inverse Functions solutions February 11, 2019 Consider vertical line test f ( x ) is a function but f 1( x ) is not a function Thus, there is no inverse function AutoSave 7 Inverse Functions solutions February 11, 2019 Fact: If the function has an inverse function then this inverse function: • can be predetermined using the horizontal line test Consider horizontal line test f(x) f ( x ) will have an inverse function as it passes the horizontal line test AutoSave 8 Inverse Functions solutions February 11, 2019 horizontal line test AutoSave 9 Inverse Functions solutions February 11, 2019 Consider functions with a restricted domain AutoSave 10 Inverse Functions solutions February 11, 2019 Consider functions with a restricted domain AutoSave 11 Inverse Functions solutions February 11, 2019 Features of a function and its inverse function Consider again the function 1. Features of f(x) and f 1(x) Features f(x) f 1(x) Slope xintercept yintercept AutoSave 12 Inverse Functions solutions Features f(x) February 11, 2019 f 1(x) Asymptote xintercept yintercept Domain Range AutoSave 13 Inverse Functions solutions February 11, 2019 Consider again the function 2. Find the equation of f 1(x) using coordinate geometry Features f(x) Slope 2 f 1(x) yintercept xintercept AutoSave 14 Inverse Functions solutions February 11, 2019 Consider again the function 2. Find the equation of f 1(x) using variable interchange AutoSave 15 Inverse Functions solutions February 11, 2019 Fact: If the function has an inverse function then this inverse function: • has the property • where 3. Check that AutoSave is the identity function OR 16 Inverse Functions solutions February 11, 2019 Consider the reciprocal function AutoSave 17 Inverse Functions solutions February 11, 2019 Summary of facts If the function has an inverse function then this inverse function: • is denoted • is the reflection of • must be a function in the line i.e. must satisfy the vertical line test • can be predetermined using the horizontal line test • has the property • where AutoSave is the identity function 18 Inverse Functions solutions Consider AutoSave . Find February 11, 2019 using variable interchange. 19 Inverse Functions solutions AutoSave February 11, 2019 20