Transformations and Properties of Functions in College Algebra

Functions: Domain, Range, and Symmetry

Determining the Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. To find the domain, consider restrictions such as division by zero or taking the square root of a negative number.

• Polynomial functions (e.g., $f(x) = 2x^2 - 5x$): Domain is all real numbers, $(-cfty, 5cfty)$.

• Rational functions (e.g., $g(x) = \frac{2x}{x-5}$): Exclude values that make the denominator zero. For $g(x)$, domain is $x \neq 5$.

• Radical functions (e.g., $h(x) = \sqrt{x^2 - x - 8}$): Set the radicand $x^2 - x - 8 \geq 0$ and solve for $x$.

• Square root functions (e.g., $f(x) = \sqrt{x-5}$): Require $x-5 \geq 0$, so domain is $x \geq 5$.

• Functions with denominators and radicals (e.g., $k(x) = \frac{\sqrt{x+2}}{x-2}$): Require $x+2 \geq 0$ and $x \neq 2$.

Example: Find the domain of $h(x) = \sqrt{x^2 - x - 8}$.

• Set $x^2 - x - 8 \geq 0$.

• Factor: $(x-4)(x+2) \geq 0$.

• Solution: $x \leq -2$ or $x \geq 4$.

Determining the Range of a Function from Its Graph

The range of a function is the set of all possible output values (y-values). To find the range from a graph, identify the lowest and highest y-values covered by the graph.

• Scan the graph from left to right to observe the y-values.

• Range is the interval from the lowest to the highest y-value.

Even, Odd, and Neither Functions

A function's symmetry can be classified as even, odd, or neither:

• Even function: $f(-x) = f(x)$ for all $x$ in the domain. The graph is symmetric about the y-axis.

• Odd function: $f(-x) = -f(x)$ for all $x$ in the domain. The graph is symmetric about the origin.

• Neither: If neither condition is met.

Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Transformations of Functions

Vertical Shifts

Vertical shifts move the graph of a function up or down.

• Upward shift: $y = f(x) + c$ shifts the graph up $c$ units.

• Downward shift: $y = f(x) - c$ shifts the graph down $c$ units.

Example: If $f(x) = |x|$, then $f(x) + 2$ shifts the graph up 2 units.

Horizontal Shifts

Horizontal shifts move the graph left or right.

• Right shift: $y = f(x - c)$ shifts the graph right $c$ units.

• Left shift: $y = f(x + c)$ shifts the graph left $c$ units.

Example: $f(x) = x^2$; $f(x-3)$ shifts the graph right 3 units.

Vertical Stretch and Compression

Multiplying a function by a constant $a$ affects its vertical stretch or compression.

• Vertical stretch: If $a > 1$, $y = a f(x)$ stretches the graph vertically by a factor of $a$.

• Vertical compression: If $0 < a < 1$, $y = a f(x)$ compresses the graph vertically by a factor of $a$.

Example: $f(x) = x^2$; $g(x) = 2x^2$ is a vertical stretch by 2.

Reflections

Reflections flip the graph over an axis.

• Reflection over the x-axis: $y = -f(x)$.

• Reflection over the y-axis: $y = f(-x)$.

Example: $f(x) = \sqrt{x}$; $g(x) = -\sqrt{x}$ is a reflection over the x-axis.

Graphing Transformations: Step-by-Step Examples

Vertical Shifts Example

x	f(x) = |x|	f(x) + 2	f(x) - 3
-2	2	4	-1
-1	1	3	-2
0	0	2	-3
1	1	3	-2
2	2	4	-1

Horizontal Shifts Example

x	f(x) = x^2	f(x+2) = (x+2)^2	f(x-3) = (x-3)^2
-2	4	0	25
-1	1	1	16
0	0	4	9
1	1	9	4
2	4	16	1

Reflection Example

• Given $f(x) = \sqrt{x}$, the graph of $y = -\sqrt{x}$ is a reflection over the x-axis.

• Given $f(x) = \sqrt{x}$, the graph of $y = \sqrt{-x}$ is a reflection over the y-axis.

Vertical Stretch/Compression Example

x	f(x) = x^2	g(x) = 2x^2	h(x) = 0.5x^2
-2	4	8	2
-1	1	2	0.5
0	0	0	0
1	1	2	0.5
2	4	8	2

Summary Table: Types of Transformations

Transformation	Equation	Effect on Graph
Vertical Shift	$y = f(x) + c$	Up $c$ units
Vertical Shift	$y = f(x) - c$	Down $c$ units
Horizontal Shift	$y = f(x - c)$	Right $c$ units
Horizontal Shift	$y = f(x + c)$	Left $c$ units
Vertical Stretch	$y = a f(x), a > 1$	Stretched vertically by $a$
Vertical Compression	$y = a f(x), 0 < a < 1$	Compressed vertically by $a$
Reflection over x-axis	$y = -f(x)$	Flipped over x-axis
Reflection over y-axis	$y = f(-x)$	Flipped over y-axis

Key Takeaways

• Always check for domain restrictions before graphing or transforming functions.

• Use tables of values to visualize transformations.

• Understand symmetry to classify functions as even, odd, or neither.

• Apply vertical and horizontal shifts, stretches, compressions, and reflections to basic graphs to obtain new functions.

Additional info: These notes expand on handwritten and tabular examples, providing full definitions, explanations, and structured tables for clarity and completeness.