warm up f(x) = x2 + 4 g(x) = 4 - x (f о g)(x)= (g о f)(x)=

Horizontal translations change both the rules and the intervals of piecewise functions. Vertical translations change only the rules. Caution

Example 1: Transforming Piecewise Functions 2 – x if x < 0 Given f(x) = write the rule g(x), a vertical translation up 3. 1 2 x2 if x ≥ 0

Example 1: Transforming Piecewise Functions 2x - 4 if x < 2 Given f(x) = write the rule g(x), a horizontal translation left 2. x2 if x ≥ 2

Intercepts x intercept replace y with zero and solve y intercept replace x with zero and solve

Find the X and Y intercepts for the piecewise function before and after the transformation 1 2 – x if x < 0 Given f(x) = write the rule g(x), a vertical translation up 3. 1 2 x2 if x ≥ 0

Find the X and Y intercepts for the piecewise function before and after the transformation 2x - 4 if x < 2 Given f(x) = write the rule g(x), a horizontal translation left 2. x2 if x ≥ 2

In mathematics, an _________ __________ is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x. Example: f(x) = x+4 [(1,5),(2,6),(3,7),(4,8)] f-1(x) = x-4 [(5,1),(6,2),(7,3),(8,4)]

To write the rule for the inverse of a function, you can exchange x and y and solve the equation for y. Because the value of x and y are switched, the domain of the function will be the range of its inverse and vice versa.

Find the inverse of . Determine whether it is a function, and state its domain and range.

Find the inverse of f(x) = x3 – 2 Find the inverse of f(x) = x3 – 2. Determine whether it is a function, and state its domain and range.

Definition of Inverse Function Let f and g be two functions such that f(g(x)) = x for every x in the domain of g g(f(x)) = x for every x in the domain of f

Find the inverse function of f(x) = 8x Then verify that both f(f-1(x)) and f-1(f(x)) are equal to the identify function

Which of these functions is the inverse of f(x) = 7x+4? g(x) = (x-4)/7 h(x) = (x-7)/4

one-to-one function a function f is ______ ___ ______ if each x value corresponds to exactly one y value. When both a relation and its inverses are functions, the relation is a one-to-one function. **They pass the horizontal and vertical line test. **

Determine by composition whether each pair of functions are inverses. f(x) = 3x – 1 and g(x) = x + 1

Determine by composition whether each pair of functions are inverses. 3 2 f(x) = x + 6 and g(x) = x – 9

Process of finding an inverse Use the horizontal line test to decide whether f has an inverse function in the equation for f(x) replace f(x) by y. interchange x and y, and solve for y verify that f(g(x))=x and g(f(x)) = x

f(x) = 3x+1

f(x) = x3+1

f(x) = x3 -1