1.5-1.6 Review
A graphical test for the existence of an inverse function is called the _______ Line Test
Horizontal
composition
f ° g (x) or g ° f (x)
(f ° g)(x) =
f(g(x)) or plug in g(x) into f(g)
(f + g)(x) =
f(x) + g(x)
(f - g)(x) =
f(x) - g(x)
(f × g)(x) =
f(x) × g(x)
Two functions f and g can be combined by the arithmetic operations of _____ _____ _____ and _____ to create new functions
(f + g)(x) (f - g)(x) (f × g)(x) (f ÷ g)(x)
inverse function
-interchanging the first and second coordinates of each of the ordered pairs -denoted as f⁻¹ (x) -graph is a reflection
algebraically solve for inverses
-write the equation -sub in x for f(x) and y for x -isolate and solve for y -rewrite
ex: write the function h(x) = (3x - 5)³
break it down into f(x) = and g(x) = **g(x) usually goes inside something; so you can do: f(x) = x³ and g(x) = 3x - 5
The _________ of a function f with g is (f ° g)(x) = f(g(x))
composite
(f ÷ g)(x) =
f(x) ÷ g(x)
domain of arithmetic combination
functions f and g consist of all real numbers that are common of f and g; f(x) ÷ g(x) can NEVER be g(x) = 0
The domain of f ° g is the set of all x in the domain of g such that ______ is the domain of f
g(x)
To decompose a composite function, look for an _____ and ________ function
inner outer
If the composite function f(g(x)) = x and g(f(x)) = x, then the function g is the ____ function of f and is denoted by ____
inverse f⁻¹
ex: sets in inverse functions: f⁻¹(x) = x−4 (5,1)(6,2)(7,3)(8,4)
is the inverse of f(x) = x + 4 (1,5)(2,6)(3,7)(4,8)
one-to-one test
make f(a) = f(b) and solve for a = b *** if you must take an even root of the numbers, then a≠b
To have an inverse function, a function f must be _______; that is f(a) = f(b) implies that a = b
one-to-one
To have an inverse function, a function must be ______, which means ...
one-to-one; no two elements in the domain of f correspond to the element in the range of f; use one-to-one test
The domain of f is ____ of f⁻¹ and the ____ of f⁻¹ is the range of f
range domain
graphs of inverse
reflection on calculator: use draw inverse (2nd DRAW)
domain and range in inverse functions
the domain of f must be equal to the range of f⁻¹ and the range of f must be equal to the domain of f⁻¹; are just flipped
horizontal line test
used to show whether a function of x is one-to-one; check a graph to see that every horizontal line interests the graph of the function at MOST one time
The graphs of f and f⁻¹ are reflections of each other in the line ______
y = x