1.3 Transformations and Combinations of Functions

Transformations of Functions

  • Translations (shifts)

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

  • Vertical and horizontal scaling and reflections

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

  • Absolute value transformation

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

  • Other example

    • Vertical stretch example: to get y=2f(x)y = 2 f(x), multiply the y-coordinates by 2.
    • Example: y=2cos⁡xy = 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)(f+g)(x) = f(x) + g(x)
    • Difference: (f−g)(x)=f(x)−g(x)(f-g)(x) = f(x) - g(x)
    • Product: (fg)(x)=f(x)⋅g(x)(fg)(x) = f(x) \cdot g(x)
    • Quotient: (fg)(x)=f(x)g(x)\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}

  • Domain considerations for combinations

    • Domain of f+gf+g and f−gf-g: dom(f±g)=dom(f)∩dom(g)\text{dom}(f\pm g) = \text{dom}(f) \cap \text{dom}(g)
    • Domain of fgfg: dom(fg)=dom(f)∩dom(g)\text{dom}(fg) = \text{dom}(f) \cap \text{dom}(g)
    • Domain of fg\frac{f}{g}: exclude points where g(x)=0g(x) = 0
    • dom(fg)=x∈dom(f)∩dom(g)∣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∘g)(x)=f(g(x))(f \circ g)(x) = f\bigl(g(x)\bigr)
    • Domain: dom(f∘g)=x∈dom(g)∣g(x)∈dom(f)\text{dom}(f \circ g) = { x \in \text{dom}(g) \mid g(x) \in \text{dom}(f) }
    • Notation: f∘g or fg!f \circ g \text{ or } f g!
    • Example with simple functions:
    • If f(x)=x2f(x) = x^2 and g(x)=x−3g(x) = x - 3, then (f∘g)(x)=(x−3)2(f \circ g)(x) = (x - 3)^2
    • (g∘f)(x)=g(f(x))=x2−3(g \circ f)(x) = g(f(x)) = x^2 - 3

  • Multi-stage composition

    • Composition of three functions: (f∘g∘h)(x)=f(g(h(x)))(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)