Precalculus Study Notes: Functions, Graphs, and Applications

3.1 Functions and Their Properties

Definition of a Function

A function is a relation in which each element of the domain (input) is paired with exactly one element of the range (output). If a domain value (x) is used more than once for different outputs, the relation is not a function.

• Example: f(x) = x^2 is a function; x = ±√y is not a function because a single y-value can correspond to two x-values.

• Domain: Set of all possible input values (x-values).

• Range: Set of all possible output values (y-values).

Difference Quotient

The difference quotient is used to compute the average rate of change of a function over an interval and is foundational for calculus.

• Formula: $\frac{f(x+h) - f(x)}{h}$, where $h \neq 0$

• Steps:

  1. Replace x with (x + h) in the function.

  2. Subtract f(x).

  3. Simplify the numerator.

  4. Factor and cancel h if possible.

• Example: For $f(x) = x^2$, $\frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$

Domain of a Function

To find the domain, identify all x-values for which the function is defined.

• Linear equations: Domain is all real numbers ($\mathbb{R}$).

• Quadratic equations: Domain is all real numbers ($\mathbb{R}$).

• Rational functions: Exclude x-values that make the denominator zero.

• Square root functions: The radicand must be non-negative ($\geq 0$).

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

• Example: $f(x) = \sqrt{x-3}$, domain is $x \geq 3$.

3.2 Graphs of Functions

Vertical Line Test

A graph represents a function if and only if no vertical line intersects the graph more than once.

• If a vertical line crosses the graph more than once, the relation is not a function.

Intercepts

• x-intercept: Set y = 0 and solve for x.

• y-intercept: Set x = 0 and solve for y.

• Example: For $y = x^2 - 4$, x-intercepts at $x = 2, -2$; y-intercept at $y = -4$.

Symmetry

• Even function: Symmetric about the y-axis. $f(-x) = f(x)$

• Odd function: Symmetric about the origin. $f(-x) = -f(x)$

Finding Points and Intercepts

• To find a point, plug in a value for x and solve for y.

• To find intercepts, set the other variable to zero and solve.

3.3 Even and Odd Functions

Functions can be classified as even, odd, or neither based on their symmetry properties.

• Even Function: $f(-x) = f(x)$ (symmetric about the y-axis)

• Odd Function: $f(-x) = -f(x)$ (symmetric about the origin)

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

3.4 Types of Functions and Piecewise Functions

Common Functions

• Linear Function: $f(x) = mx + b$

• Quadratic Function: $f(x) = ax^2 + bx + c$

• Cubic Function: $f(x) = ax^3 + bx^2 + cx + d$

• Square Root Function: $f(x) = \sqrt{x}$

• Reciprocal Function: $f(x) = \frac{1}{x}$

• Absolute Value Function: $f(x) = |x|$

• Greatest Integer Function: $f(x) = \lfloor x \rfloor$

Piecewise Defined Functions

A piecewise function is defined by different expressions over different intervals of the domain.

• Specify the function rule for each interval.

• To find intercepts, plug in values as appropriate for each piece.

3.5 Transformations of Functions

Vertical and Horizontal Shifts

• Vertical Shift: $f(x) + k$ shifts up if $k > 0$, down if $k < 0$.

• Horizontal Shift: $f(x + h)$ shifts left if $h > 0$, right if $h < 0$.

Reflections

• Across x-axis: $-f(x)$

• Across y-axis: $f(-x)$

Stretches and Compressions

• Vertical Stretch/Compression: $a f(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

• Horizontal Stretch/Compression: $f(bx)$ compresses if $|b| > 1$, stretches if $0 < |b| < 1$.

3.16 Geometry Applications

Distance Formula

• Distance between $(x_1, y_1)$ and $(x_2, y_2)$: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Area and Perimeter

• Rectangle: Area = $l \times w$, Perimeter = $2l + 2w$

• Triangle: Area = $\frac{1}{2} b h$

• Circle: Area = $\pi r^2$, Circumference = $2\pi r$

Pythagorean Theorem

• $a^2 + b^2 = c^2$ for right triangles

4.4 Revenue and Maximization

Revenue Function

• Revenue: $R = x p$ where x = quantity sold, p = price per unit

• To maximize revenue, express R as a quadratic function and use the vertex formula.

• Vertex Formula: For $ax^2 + bx + c$, the maximum (or minimum) occurs at $x = -\frac{b}{2a}$

• Example: If $R = (100,000 - 1000x)x$, expand to $R = 100,000x - 1000x^2$ and find the maximum using the vertex formula.

Standard Form of Quadratic

• Expand and simplify expressions to write quadratics in standard form: $ax^2 + bx + c$

• Example: $2(x-5)(x+3) + 3 = 2(x^2 - 2x - 15) + 3 = 2x^2 - 4x - 30 + 3 = 2x^2 - 4x - 27$