1.3 Transformations and Combinations of Functions

Transformations of Functions

• Translations (shifts)

  • Upward shift by c (c > 0): y = f(x) + c
  • Right shift by c (c > 0): y = f(x - c)
  • Left shift by c (c > 0): y = f(x + c)
  • Downward shift by c (c > 0): y = f(x) - c

• Vertical and horizontal scaling and reflections

  • Vertical stretch by factor c: y = c f(x)
  • Horizontal stretch by factor c: y = f\left(\frac{x}{c}\right)
  • Horizontal compression by factor c: y = f(cx)
  • Reflection about axes:
  • About the x-axis: y = -f(x)
  • About the y-axis: y = f(-x)

• Absolute value transformation

  • y = |f(x)|: portion above x-axis remains; below is reflected across the x-axis

• Other example

  • Vertical stretch example: to get y = 2 f(x), multiply the y-coordinates by 2.
  • Example: y = 2\cos x is a vertical stretch of the cosine graph by a factor of 2.

Combinations of Functions

• Basic two-function combinations (f and g)

  • Sum: (f+g)(x) = f(x) + g(x)
  • Difference: (f-g)(x) = f(x) - g(x)
  • Product: (fg)(x) = f(x) \cdot g(x)
  • Quotient: \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}

• Domain considerations for combinations

  • Domain of f+g and f-g: \text{dom}(f\pm g) = \text{dom}(f) \cap \text{dom}(g)
  • Domain of fg: \text{dom}(fg) = \text{dom}(f) \cap \text{dom}(g)
  • Domain of \frac{f}{g}: exclude points where g(x) = 0
  • \text{dom}\left(\frac{f}{g}\right) = { x \in \text{dom}(f) \cap \text{dom}(g) \mid g(x) \neq 0 }

• Composition of functions

  • Definition: (f \circ g)(x) = f\bigl(g(x)\bigr)
  • Domain: \text{dom}(f \circ g) = { x \in \text{dom}(g) \mid g(x) \in \text{dom}(f) }
  • Notation: f \circ g \text{ or } f g!
  • Example with simple functions:
  • If f(x) = x^2 and g(x) = x - 3 , then (f \circ g)(x) = (x - 3)^2
  • (g \circ f)(x) = g(f(x)) = x^2 - 3

• Multi-stage composition

  • Composition of three functions: (f \circ g \circ h)(x) = f\bigl(g(h(x))\bigr)
  • Order: apply h first, then g, then f (machine interpretation: g after h, then f after g)