Lecture Notes: Transformations, Symmetry, Operations on Functions, Difference Quotient, and Composite Functions

Transformations of Absolute Value and Related Functions

• Base function: $f(x)=|x|$
• Example 1: $h(x)=-|x+3|+1$
  • Interpretation of transformations:
  • Inside absolute value: $x+3\Rightarrow$ shift left 3 units.
  • Outside: $-1\times|\cdot|$ reflects across the x-axis (inversion).
  • Finally: $+1$ shifts the graph up by 1.
  • End result described: a graph that is shifted left 3, reflected about the x-axis, and moved up by 1.
  • Quick check: you can also verify by plotting ordered pairs or using a table.
• Example 2: $h(x)=|2x-4|$
  • Start from $|x|$ and analyze the inside:
  • Factor inside: $|2x-4|=|2(x-2)|=2|x-2|.$
  • Interpretations:
  • Horizontal shift: $x-2$ indicates shift right by 2.
  • Vertical stretch: the factor 2 in front indicates a vertical stretch by a factor of 2.
  • End result: the graph of $|x|$ shifted right by 2 and stretched vertically by 2.
• Example 3: $g(x)=-\tfrac{1}{2}x^{2}$
  • Start with the familiar parabola $f(x)=x^{2}$.
  • Transformations applied:
  • Multiply by $\tfrac{1}{2}$: vertical compression by a factor of 1/2 (i.e., the parabola gets narrower).
  • Multiply by $-1$: reflection about the x-axis (opens downward).
  • Add 4: shift upward by 4.
  • Resulting graph: a downward-opening parabola (due to the negative) shifted up by 4 units.
• Distinguishing between vertical vs horizontal translations:
  • If you have $f(x+3)$, this is a horizontal shift left by 3.
  • If you have $f(x)+3$, this is a vertical shift up by 3.
  • If you combine them as $f(x-2)+3$, you get a right shift of 2 and an up shift of 3; order does not affect the final vertical/horizontal amounts, but you must interpret each part correctly.
• Quick practical note on using translations and transformations:
  • The translations, reflections, and stretches can be understood via the inside and outside of the function notation.
  • This approach lets you infer the graph quickly without plotting every point.

Symmetry of Graphs and Functions

• General idea: symmetry about an axis or the origin reveals fundamental invariances of the graph.
• Symmetry about the y-axis (vertical axis):
  • Test: replace $x$ by $-x$ in the equation; if the equation is equivalent, the graph is symmetric about the y-axis.
  • For a function $f(x)$ in the form $y=f(x)$, this means $f(-x)=f(x)$ for all x in the domain.
  • Example: $y=x^{2}+4$ is symmetric about the y-axis because $(-x)^{2}+4=x^{2}+4$.
• Symmetry about the x-axis (horizontal axis):
  • Test: replace $y$ by $-y$; if the equation remains equivalent, the graph is symmetric about the x-axis.
  • For $y=f(x)$, this generally is not true unless the graph satisfies $f(x)=-f(x)$ for all x (which implies $f(x)=0$ for all x, a degenerate case).
  • Example: the circle $x^{2}+y^{2}=16$ is symmetric about both axes (and about the origin) because it remains invariant under the indicated transformations.
• Symmetry about the origin:
  • Test: replace (x,y) with (-x,-y); if equivalent, the graph is symmetric about the origin.
  • For a relation $y=f(x)$, origin symmetry occurs if $f(-x)=-f(x)$ (the function is odd).
  • Examples:
  • The circle $x^{2}+y^{2}=16$ is origin-symmetric when centered at the origin.
  • The curve $y=x^{3}$ is origin-symmetric because $(-x)^{3}=-x^{3}$.
  • The relation $y=\frac{1}{x}$ is origin-symmetric as well (since replacing gives equivalent form).
• Quick practice checks (picked from the lecture):
  • Test various equations to determine their axis/origin symmetry using the appropriate substitutions.
  • If graphed, symmetry is often visually apparent; the tests help when graphs aren’t available.

Operations on Functions and Their Domains

• Notation to combine two functions, defined where both are defined:
  • $(f+g)(x) = f(x) + g(x)$
  • $(f-g)(x) = f(x) - g(x)$
  • $(f\cdot g)(x) = f(x)\,g(x)$
  • $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \quad \text{where } g(x) \neq 0$
• Intuition:
  • These are simply the pointwise addition, subtraction, multiplication, and division of the two functions at each x.
  • The domain of the resulting function is restricted by where both f and g are defined (and additionally where g(x) ≠ 0 for the quotient).
• Worked idea from the lecture (conceptual steps):
  • To evaluate (f+g)(1), compute f(1) and g(1), then add them. The same approach applies to f-g, f·g, and f/g (with the domain caveat for division).
• Example prompts (as discussed in class):
  • If given specific f and g, you would compute values like f(1) + g(1), f(-3) - g(-3), etc., and verify with algebra as needed.
• Quick takeaway:
  • This notation emphasizes the modular combination of two known functions without needing to rewrite each time.

The Difference Quotient

• Definition and purpose:
  • The difference quotient is the core building block for the derivative; it measures the average rate of change of f over an interval of length $h$.
  • Formula:
    $DQ_f(x,h) = \frac{f(x+h) - f(x)}{h}, \quad h \neq 0.$
• Important restriction: $h\neq 0$ because division by zero is undefined; the limit as $h\to 0$ gives the instantaneous rate of change.
• Example: take $f(x)=2x^{2}-3$.
  • Compute inner value: $f(x+h)=2(x+h)^{2}-3 = 2x^{2}+4xh+2h^{2}-3.$
  • Difference: $f(x+h)-f(x) = (2x^{2}+4xh+2h^{2}-3) - (2x^{2}-3) = 4xh+2h^{2}.$
  • Difference quotient: $DQ_f(x,h) = \frac{4xh+2h^{2}}{h} = 4x+2h.$
• Observations:
  • The algebra should cancel like terms to simplify the expression.
  • If you take the limit as $h\to 0$, you obtain the derivative: $f'(x) = \lim<em>{h\to 0} DQ</em>f(x,h) = 4x.$
• Practical tip:
  • Carefully expand and collect like terms; check cancellations (e.g., symmetric terms like $2x^{2}$ cancel when subtracting f(x) from f(x+h)).

Composite Functions

• Definition:
  • The composition of f and g is the function $(f\circ g)(x) = f\big( g(x) \big)$
  • Conceptually: a function inside a function; you feed the output of g into f.
• Important: the order matters; in general, $(f\circ g)(x) \neq (g\circ f)(x)$.
• Worked example from the lecture:
  • Given $f(x) = 2x-1$ and $g(x) = \dfrac{4}{x-1}$
  • Compute $(f\circ g)(2) = f\big( g(2) \big) <br /> = f(4) = 2\cdot 4 - 1 = 7.$
  • Compute $(g\circ f)(2) = g\big( f(2) \big) <br /> = g(3) = \dfrac{4}{3-1} = 2.$
• Another check (partial): for x = -3, with the same f and g,
  • $f(-3) = 2(-3) - 1 = -7.$
  • Then $(g\circ f)(-3) = g(-7) = \dfrac{4}{-7-1} = \dfrac{4}{-8} = -\tfrac{1}{2}.$
• Takeaway:
  • Always start with the inner function, evaluate it, then plug that result into the outer function.
  • Changing the order changes the result, illustrating the non-commutativity of function composition.

Connections to Foundational Concepts and Applications

• The translation, reflection, and scaling rules connect to the geometric understanding of graphs, which in turn underpin modeling in physics, engineering, and economics.
• Symmetry is a powerful diagnostic tool in problem solving, reducing the complexity of solving equations or understanding the behavior of systems.
• The difference quotient introduces the fundamental idea of instantaneous rate of change, leading directly to derivatives and optimization.
• Composite functions are central to modeling layered processes (e.g., applying a transformation G to data, then applying a decision rule F to the result).

Quick Recap of Key Formulas

• Absolute value transformations:
  • $h(x) = a\,|b x + c| + d$
  • Example interpretations:
  • Inside term shifts left/right: $x+c$ or scaling via $b$ inside absolute value.
  • Outside amplitude/vertical change: multiply by $a$ and shift by $d$.
• Symmetry tests:
  • Y-axis: $f(-x) = f(x) \text{for all } x$
  • X-axis: replace $y$ with $-y$ in the equation (invariance criterion)
  • Origin: replace $(x,y)$ with $(-x,-y)$ (inference: odd symmetry: $f(-x)=-f(x)$)
• Operations on functions:
  • $(f\pm g)(x) = f(x) \pm g(x)$
  • $(f\cdot g)(x) = f(x)\,g(x)$
  • $\left(\frac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}, \quad g(x) \neq 0$
• Difference quotient:
  • $DQ_f(x,h) = \dfrac{f(x+h) - f(x)}{h}, \quad h \neq 0$
  • Example: with $f(x)=2x^{2}-3$, $f(x+h)=2(x+h)^{2}-3$ and
    $DQ_f(x,h)=\dfrac{(2x^{2}+4xh+2h^{2}-3) - (2x^{2}-3)}{h} = 4x+2h$
• Composite functions:
  • $(f\circ g)(x) = f(g(x))$
  • Examples: with $f(x)=2x-1, \ g(x)=\frac{4}{x-1}$
  • $(f\circ g)(2) = 7$
  • $(g\circ f)(2) = 2$
  • $(g\circ f)(-3) = -\tfrac{1}{2}$