lecture 8: Operations on functions — Transformations: shifting, squeezing, reflecting, and composing functions

1. Motivation: Why Transform and Compose?

In modeling real‐world phenomena, we often need to tweak basic functions to fit data or combine them to capture complex relationships:

• Audio volume control: stretching or squeezing the waveform.
• Image translation: shifting pixel coordinates.
• Signal inversion: reflecting a waveform.
• Nested processes: feeding the output of one function into another (composition).

2. Transformations of the Graph

A transformation modifies the graph of a “parent” function $f(x)$.

2.1 Shifting

• Vertical shift: $g(x)=f(x)+k$
  • Up $k>0$, down $k<0$.
  • Example: $g(x)=x^2+3$ moves the parabola up 3 units.
• Horizontal shift: $g(x)=f(x-h)$
  • Right $h>0$, left $h<0$.
  • Example: $g(x)=(x-2)^2$ moves $f(x)=x^2$ right 2 units.

Graphical summary

2.2 Stretching and Squeezing

• Vertical stretch/compress: $g(x)=a\,f(x)$
  • Stretch if $|a|>1$, compress if $0<|a|<1$.
  • Example: $2\sin x$ is a vertical stretch of $\sin x$ by factor 2.
• Horizontal stretch/compress: $g(x)=f(bx)$
  • Compress if $|b|>1$, stretch if $0<|b|<1$.
  • Example: $\sin(2x)$ is horizontally compressed by factor $\tfrac12$.

2.3 Reflection

• Reflect about the x-axis: $g(x)=-f(x)$
  • Flips graph upside down.
• Reflect about the y-axis: $g(x)=f(-x)$
  • Mirrors graph left to right.

Example:
Reflect $f(x)=2^x$ about the y-axis to get $g(x)=2^{-x}$, which decays instead of grows.

3. Combining Transformations

Transformations can be chained. Always apply inside-the-function (horizontal) shifts/scales first, then reflections, then outside (vertical) scales/shifts.

Example:
Given
$f(x)=x^2,\quad g(x)=-2\,f(3(x+1)) + 4$

1. Inside: $x\to 3(x+1)$ → shift left 1, compress horizontally by 1/3.
2. Reflect x-axis and stretch vertically by 2.
3. Shift up 4.

4. Composition of Functions

Composition feeds one function into another:

$$(h\circ g)(x) = h\bigl(g(x)\bigr).$$

• Notation: $h\circ g$ or $h(g(x))$.
• Domain: those $x$ for which $g(x)$ is in the domain of $h$.

Real-world example:
Temperature conversion then rate calculation:

$$C(t)=\tfrac{5}{9}(F(t)-32), \quad R(C)=2C+1 \quad (R\circ C)(t)=2\bigl(\tfrac{5}{9}(F(t)-32)\bigr)+1$$

4.1 Decomposing a Function

Given $h(x)=\sqrt{2x+3}$, one composition is:

• $g(x)=2x+3$
• $f(u)=\sqrt{u}$
• Then $h=f\circ g$.

You can often find multiple decompositions by regrouping operations.

Regular Exercises

1. Graphing Transformations

Sketch $f(x)=\log_{2}(x)$ and then sketch $g(x)=-3\,\log_{2}(2(x-1))+2$. Label asymptotes and intercepts.

2. Chain of Transformations

Describe the sequence of shifts, stretches, and reflections to go
from $y=x^3$ to $y=4\,( - (x+2)^3 ) -1$.

3. Composition Practice

If $f(x)=x^2-1$ and $g(x)=3x+2$, find and simplify $(f\circ g)(x)$ and $(g\circ f)(x)$. Determine their domains.

4. Transformation Proof

Let $f(x)=x^2, k=2, h=3$.
Prove that performing a horizontal stretch by factor $k$ then a shift by $h$ is not the same as shifting first then stretching (unless $h=0$).

5. Domain Composition

Given $f(x)=\sqrt{x-1}$ and $g(x)=\log_{2}(x)$, determine the domain of $g\circ f$ and $f\circ g$, and explain any restrictions.