College Algebra Study Guide: Piecewise Functions, Function Operations, Transformations, and Variation

Piecewise Functions

Definition and Graphing

A piecewise function is defined by different expressions depending on the input value (domain). Each 'piece' applies to a specific interval of the independent variable.

• Definition: A function composed of two or more sub-functions, each with its own domain.

• Example: $f(x) = \begin{cases} -4 & \text{for } x \leq -1 \\ 6 & \text{for } x > -1 \end{cases}$

• Graphing: Plot each piece on its respective interval, using open or closed circles to indicate whether endpoints are included.

Evaluating Piecewise Functions

To find $f(a)$, determine which interval $a$ falls into and use the corresponding expression.

• Example: For $f(x)$ above, $f(-2) = -4$ (since $-2 \leq -1$), $f(5) = 6$ (since $5 > -1$).

Intervals of Increase, Decrease, and Constancy

Analyze each piece to determine where the function is increasing, decreasing, or constant.

• Increasing: Function values rise as $x$ increases.

• Decreasing: Function values fall as $x$ increases.

• Constant: Function values remain unchanged as $x$ increases.

• Example: For $f(x)$ above, both pieces are constant on their intervals.

Function Operations and Composition

Basic Operations

Functions can be added, subtracted, multiplied, or divided to create new functions.

• Addition: $(f+g)(x) = f(x) + g(x)$

• Subtraction: $(f-g)(x) = f(x) - g(x)$

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Function Composition

The composition of functions, denoted $(f \circ g)(x)$ or $f(g(x))$, means applying $g$ first, then $f$ to the result.

• Example: If $f(x) = 3x + 2$ and $g(x) = 5x - 4$, then $(f \circ g)(x) = f(g(x)) = 3(5x - 4) + 2 = 15x - 12 + 2 = 15x - 10$

Domain of Combined Functions

The domain of a combined function is the set of all $x$ values for which the operation is defined.

• For addition, subtraction, multiplication: Intersection of the domains of $f$ and $g$.

• For division: Intersection of domains, excluding values where $g(x) = 0$.

• For composition: $x$ must be in the domain of $g$, and $g(x)$ must be in the domain of $f$.

Transformations of Functions

Types of Transformations

Transformations change the position or shape of a function's graph.

• Vertical Shifts: $g(x) = f(x) + k$ shifts the graph up ($k > 0$) or down ($k < 0$).

• Horizontal Shifts: $g(x) = f(x - h)$ shifts the graph right ($h > 0$) or left ($h < 0$).

• Reflections: $g(x) = -f(x)$ reflects across the $x$-axis; $g(x) = f(-x)$ reflects across the $y$-axis.

• Vertical Stretch/Compression: $g(x) = a f(x)$ stretches ($|a| > 1$) or compresses ($0 < |a| < 1$) vertically.

Examples of Transformations

• Quadratic: $f(x) = x^2$, $g(x) = x^2 + 4$ (vertical shift up by 4)

• Square Root: $f(x) = \sqrt{x}$, $g(x) = -\sqrt{x} - 2$ (reflection over $x$-axis, shift down by 2)

• Reciprocal: $f(x) = \frac{1}{x}$, $g(x) = \frac{1}{x} - 3$ (vertical shift down by 3)

• Absolute Value: $f(x) = |x|$, $g(x) = 3|x + 2| - 1$ (horizontal shift left by 2, vertical stretch by 3, shift down by 1)

Variation Problems

Direct and Inverse Variation

Variation describes how one quantity changes in relation to another.

• Direct Variation: $y = kx$ (as $x$ increases, $y$ increases proportionally)

• Inverse Variation: $y = \frac{k}{x}$ (as $x$ increases, $y$ decreases proportionally)

• Variation with Powers: $y$ varies inversely as the square of $x$: $y = \frac{k}{x^2}$

• Variation with Roots: $y$ varies directly as the square root of $x$: $y = k\sqrt{x}$

Solving Variation Problems

• Find the constant $k$: Substitute given values into the variation equation and solve for $k$.

• Use $k$ to solve for unknowns: Substitute $k$ and the new value of $x$ to find $y$.

• Example: If $y$ varies inversely as $x^2$, and $y = 0.15$ when $x = 0.1$, then $0.15 = \frac{k}{(0.1)^2}$, so $k = 0.15 \times 0.01 = 0.0015$.

Summary Table: Types of Function Transformations

Transformation	Equation	Effect on Graph
Vertical Shift	$g(x) = f(x) + k$	Up if $k > 0$, down if $k < 0$
Horizontal Shift	$g(x) = f(x - h)$	Right if $h > 0$, left if $h < 0$
Reflection over $x$-axis	$g(x) = -f(x)$	Flips graph vertically
Reflection over $y$-axis	$g(x) = f(-x)$	Flips graph horizontally
Vertical Stretch/Compression	$g(x) = a f(x)$	Stretches if $|a| > 1$, compresses if $0 < |a| < 1$

Additional info: The study notes above expand on the original questions by providing definitions, formulas, and examples for each concept, ensuring a self-contained guide for College Algebra students.