Study Guide: Transformations and Algebra of Functions in Precalculus

Section 3.4: Transformations of Functions

Determining the Domain of a Function

Understanding the domain of a function is essential for analyzing its behavior. The domain is the set of all possible input values (x-values) for which the function is defined.

• Polynomial functions (e.g., $f(x) = 2x^2 - 5x$) have a domain of all real numbers.

• Rational functions (e.g., $g(x) = \frac{x}{x^2 - x - 6}$) are undefined where the denominator is zero.

• Radical functions (e.g., $h(x) = \sqrt{8 - 2x - 8}$) are defined only when the expression under the square root is non-negative.

• Composite expressions may combine these restrictions.

Example: Find the domain of $f(x) = \sqrt{x - 9}$.

• Set $x - 9 \geq 0 \implies x \geq 9$.

• Domain: $[9, \infty)$

Even and Odd Functions

A function is even if $f(-x) = f(x)$ for all $x$ in the domain, and its graph is symmetric about the y-axis. A function is odd if $f(-x) = -f(x)$, and its graph is symmetric about the origin.

• Even Example: $f(x) = x^2$

• Odd Example: $f(x) = x^3$

Vertical Shifts

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

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

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

Example: $f(x) = x^2$; $f(x) + 3$ shifts the parabola up by 3 units.

Horizontal Shifts

Horizontal shifts move the graph left or right.

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

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

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

Reflections

Reflections flip the graph over an axis.

• $y = -f(x)$ reflects the graph over the x-axis.

• $y = f(-x)$ reflects the graph over the y-axis.

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

Vertical and Horizontal Stretching/Shrinking

• Vertical Stretch/Shrink: $y = a f(x)$

  • If $a > 1$, the graph is stretched vertically.

  • If $0 < a < 1$, the graph is shrunk vertically.

• Horizontal Stretch/Shrink: $y = f(ax)$

  • If $a > 1$, the graph is shrunk horizontally.

  • If $0 < a < 1$, the graph is stretched horizontally.

Example: $f(x) = |x|$; $f(2x)$ is a horizontal shrink by a factor of 2.

Order of Transformations

When multiple transformations are applied, the typical order is:

1. Horizontal shifts

2. Horizontal stretches/shrinks and reflections

3. Vertical stretches/shrinks and reflections

4. Vertical shifts

Example: For $y = -2f(x + 3) - 4$, apply the horizontal shift, then vertical stretch/reflection, then vertical shift.

Section 3.5: The Algebra of Functions; Composite Functions

Compound and Polynomial Inequalities

Solving inequalities is a key skill for determining domains and ranges.

• Compound inequalities involve 'and' ($\cap$) or 'or' ($\cup$) statements.

• Polynomial inequalities require finding intervals where the polynomial is positive or negative.

Example: Solve $x^3 - 3x^2 + 2x \geq 0$ by factoring and testing intervals.

Algebra of Functions

Functions can be combined using addition, subtraction, multiplication, and division.

• Sum: $(f + g)(x) = f(x) + g(x)$

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

• Product: $(fg)(x) = f(x) \cdot g(x)$

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

Example: If $f(x) = x^2$, $g(x) = \sqrt{x}$, then $(f + g)(x) = x^2 + \sqrt{x}$.

Composite Functions

The composition of functions, denoted $(f \circ g)(x)$, means $f(g(x))$.

• To find the domain of $f \circ g$, $x$ must be in the domain of $g$ and $g(x)$ must be in the domain of $f$.

Example: If $f(x) = \sqrt{x}$ and $g(x) = x - 1$, then $(f \circ g)(x) = \sqrt{x - 1}$, so $x - 1 \geq 0 \implies x \geq 1$.

Intersection of Intervals

To find the domain of combined functions, determine the intersection of the domains of the individual functions.

• Interval notation is used to express domains.

Example: $[0, 5) \cap (2, \infty) = (2, 5)$

Section 3.6: One-to-One Functions; Inverse Functions

One-to-One Functions

A function is one-to-one if each output value corresponds to exactly one input value. This property is necessary for a function to have an inverse.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function is not one-to-one.

Example: $f(x) = x^3$ is one-to-one; $f(x) = x^2$ is not.

Inverse Functions

The inverse of a function $f$, denoted $f^{-1}$, reverses the effect of $f$.

• If $f(a) = b$, then $f^{-1}(b) = a$.

• The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$.

Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x - 3}{2}$.

Applications: Temperature Conversion

Functions can model real-world relationships, such as temperature conversion between Celsius and Fahrenheit.

• $F(C) = \frac{9}{5}C + 32$ converts Celsius to Fahrenheit.

• $C(F) = \frac{5}{9}(F - 32)$ converts Fahrenheit to Celsius.

Example: $C(212) = \frac{5}{9}(212 - 32) = 100$ (boiling point of water).

Summary Table: Types of Function Transformations

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

Key Definitions

• Domain: Set of all input values for which a function is defined.

• Range: Set of all possible output values of a function.

• Composite Function: $(f \circ g)(x) = f(g(x))$

• Inverse Function: $f^{-1}(x)$ such that $f(f^{-1}(x)) = x$

• One-to-One Function: A function where each output is paired with exactly one input.

Examples and Applications

• Domain Example: $f(x) = \frac{1}{x-2}$ has domain $x \neq 2$.

• Transformation Example: $f(x) = |x|$, $g(x) = 3|x|$ is a vertical stretch by 3.

• Composite Example: $f(x) = x^2$, $g(x) = x+1$, $(f \circ g)(x) = (x+1)^2$.

• Inverse Example: $f(x) = 2x + 1$, $f^{-1}(x) = \frac{x-1}{2}$.

Additional info: This study guide is based on Precalculus notes covering function transformations, algebra of functions, composite and inverse functions, and related domain/range concepts. It is suitable for exam preparation and review.