Section 1.3 New Functions from Old
Multiplying a function by a constant stretches the graph vertically:
g(x) = A f (x)
Multiplying the variable x by a constant stretches the graph horizontally:
g(x) = f (Bx)
!
Adding a constant to the function shifts the graph vertically
g(x) = f (x) + k
!
Adding a constant to the variable x shifts the graph horizontally
g(x) = f (x " h)
!
Multiplying the function by (-1) reflects the graph over the x-axis
g(x) = " f (x)
!
Multiplying the variable x by (-1) reflects the graph over the y-axis
g(x) = f ("x)
!
Composite Functions: ( f o g)(x) = f (g(x))
!
Odd and Even Functions:
• A function f is even if f ("x) = f (x) for all x. An even function is symmetric with
!
respect to the y-axis.
• A function f is odd if f ("x) = " f (x) for all x. An odd function is symmetric with
respect to the origin.
!
!
Inverse functions:
! the inverse of a function we reverse the roles of input and output. In
• When finding
other words, we switch x and y.
• f "1 (y) = x means f (x) = y
• A function has a well-defined inverse function only if f(x) is one-to-one (passes
the horizontal line test)
• The graph of f-1 is the reflection of f about the line y = x
Section 1.3
Exercises: 1-13 (odd), 14, 15, 27, 29, 31-37 (odd)
Problems: 43, 45, 49, 54.