# Operations on Functions

## Contents

#### Reflections and Rotations

We can also reflect the graph of a function over the x -axis ( y = 0 ), the y -axis( x = 0 ), or the line y = x .

Making the output negative reflects the graph over the x -axis, or the line y = 0 . Here are the graphs of y = f (x) and y = - f (x) . Note that if (x, y 1) is a point on the graph of f (x) and (x, y 2) is a point on the graph of - f (x) , then y 2 = - y 1 . The range also becomes negative; if range = f (x):a < f (x) < b , then the new range is g(x):-b < g(x) < -a .

Graphs of f (x) and - f (x)

Making the input negative reflects the graph over the y -axis, or the line x = 0 . Here are the graphs of y = f (x) and y = f (- x) . Note that if (x 1, y) is a point on the graph of f (x) and (x 2, y) is a point on the graph of f (- x) , then x 2 = - x 1 . The domain also becomes negative; if domain = x:a < x < b , then the new domain is x:-b < x < -a .

Graphs of f (x) and f (- x)

Switching x and y reflects the graph over the line y = x (this is equivalent to finding the inverse). Now, x is a function of y . Here are the graphs of y = f (x) and x = f (y) . Note that if (x 1, y 1) is a point on the graph of f (x) and (x 2, y 2) is a point on the graph of f (y) , then y 1 = x 2 and y 2 = x 1 .

Graphs of y = f (x) and x = f (y)

In addition, we can rotate the graph of a function 180 o around the origin. Making the input and the output of a function negative results in a rotation of the function around the origin. Here is a graph of y = f (x) and y = - f (- x) . Note that if (x, y) is a point on the graph of f (x) , then (- x, - y) is a point on the graph of - f (- x) . The domain and range both become negative.

Graphs of f (x) and - f (- x)