Precalculus Exam 1 Study Guide: Functions, Polynomials, and Algebraic Concepts

Functions and Their Properties

Relations, Functions, Domain, and Range

A relation is a set of ordered pairs. A function is a relation in which each input (domain value) corresponds to exactly one output (range value).

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

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

• Determining a Function: A relation is a function if no input value is paired with more than one output value.

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Example: The set {(1,2), (2,3), (3,4)} is a function. The set {(1,2), (1,3)} is not a function.

Equations Defining Functions

• Given an equation in x and y, y is a function of x if for every x in the domain, there is only one corresponding y.

• To check, solve for y in terms of x and see if each x yields a unique y.

Example: $y = x^2$ defines y as a function of x. $x = y^2$ does not, since each x > 0 has two possible y values.

Evaluating Functions and Piecewise Functions

• To evaluate a function, substitute the given value into the function's formula.

• Piecewise functions are defined by different expressions over different intervals of the domain.

Example: For $f(x) = \begin{cases} x^2 & x < 0 \\ x+1 & x \geq 0 \end{cases}$, $f(-2) = 4$, $f(1) = 2$.

Increasing/Decreasing Intervals and Extrema

• A function is increasing on an interval if its output rises as x increases.

• A function is decreasing on an interval if its output falls as x increases.

• Relative minimum/maximum: The lowest/highest point in a local region of the graph.

Example: The vertex of $y = x^2$ is a relative (and absolute) minimum at (0,0).

Even and Odd Functions

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

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

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

Difference Quotient

• The difference quotient is used to compute the average rate of change of a function:

$\frac{f(x+h) - f(x)}{h}$

Transformations of Functions

• Common transformations include shifts, reflections, stretches, and compressions.

• To graph related functions, apply transformations to the graph of a basic function.

Example: $f(x) = (x-2)^2 + 3$ is $y = x^2$ shifted right 2 units and up 3 units.

Operations with Functions

• Given functions f and g, their sum, difference, product, and quotient are defined as:

$$(f+g)(x) = f(x) + g(x)$$(f-g)(x) = f(x) - g(x)$$ $$(fg)(x) = f(x) \cdot g(x)$$ $$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}, \quad g(x) \neq 0$$

• The domain of each operation is the intersection of the domains of f and g (and for the quotient, where $g(x) \neq 0$).

Composition of Functions

• The composition $f \circ g$ is defined as $f(g(x))$.

• The domain of $f \circ g$ consists of all x in the domain of g such that $g(x)$ is in the domain of f.

Example: If $f(x) = x^2$ and $g(x) = x+1$, then $(f \circ g)(x) = f(g(x)) = (x+1)^2$.

Inverse Functions

• Two functions f and g are inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all x in their domains.

• The horizontal line test determines if a function has an inverse: if every horizontal line intersects the graph at most once, the function is one-to-one and has an inverse.

• To find the inverse, solve $y = f(x)$ for x in terms of y, then interchange x and y.

Example: $f(x) = 2x + 3$ has inverse $f^{-1}(x) = \frac{x-3}{2}$.

Complex Numbers and Quadratic Equations

Operations with Complex Numbers

• Complex numbers are of the form $a + bi$, where $i = \sqrt{-1}$.

• Sum: $(a+bi) + (c+di) = (a+c) + (b+d)i$

• Difference: $(a+bi) - (c+di) = (a-c) + (b-d)i$

• Product: $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$

• Quotient: $\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)}$

Square Roots of Negative Numbers

• $\sqrt{-a} = i\sqrt{a}$ for $a > 0$.

• Operations follow the rules of complex numbers.

Solving Quadratic Equations

• Quadratic equations have the form $ax^2 + bx + c = 0$.

• Factoring: Express as $(x-p)(x-q) = 0$ and solve for x.

• Quadratic Formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Vertex and Intercepts of a Parabola

• The vertex of $y = ax^2 + bx + c$ is at $x = -\frac{b}{2a}$.

• y-intercept: Set $x = 0$.

• x-intercepts: Solve $ax^2 + bx + c = 0$.

Example: For $y = x^2 - 4x + 3$, vertex at $(2, -1)$, y-intercept at $(0,3)$, x-intercepts at $(1,0)$ and $(3,0)$.

Quadratic Functions from Conditions

• Given points or properties, set up equations to solve for coefficients a, b, c in $y = ax^2 + bx + c$.

Optimization Problems

• Model the situation with a quadratic function and find the maximum or minimum value (vertex).

Example: Maximizing area or minimizing cost using quadratic models.

Polynomial Functions and Their Properties

End Behavior of Polynomials

• As $x \to \infty$ or $x \to -\infty$, the leading term determines the end behavior.

• For $f(x) = a_nx^n + ...$, if n is even and $a_n > 0$, both ends up; if n is odd and $a_n > 0$, left down, right up.

Zeros of Polynomials and Test Points

• Find zeros by solving $f(x) = 0$.

• Use test points between zeros to determine if the graph is above or below the x-axis.

Linear Factorization Theorem

• Every polynomial of degree n can be factored as $f(x) = a_n(x - r_1)(x - r_2)...(x - r_n)$, where $r_i$ are zeros (real or complex).

Polynomial Division

• Long division and synthetic division are methods to divide polynomials.

• Quotient and remainder: $f(x) = (x - c)q(x) + r$.

Remainder and Factor Theorems

• Remainder Theorem: The remainder of $f(x)$ divided by $x - c$ is $f(c)$.

• Factor Theorem: $x - c$ is a factor of $f(x)$ if and only if $f(c) = 0$.

Rational Root Theorem

• Possible rational roots of $f(x) = a_nx^n + ... + a_0$ are $\pm \frac{p}{q}$, where p divides $a_0$ and q divides $a_n$.

Solving Polynomial Equations

• Find one or more zeros, factor, and solve for all roots.

Summary Table: Key Concepts and Theorems

Concept	Definition/Method	Key Formula
Function	Each input has one output	-
Domain	All possible x-values	-
Range	All possible y-values	-
Difference Quotient	Average rate of change	$\frac{f(x+h) - f(x)}{h}$
Quadratic Formula	Solves $ax^2 + bx + c = 0$	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Vertex of Parabola	Maximum or minimum point	$x = -\frac{b}{2a}$
Remainder Theorem	Remainder of division by $x-c$	$f(c)$
Factor Theorem	$x-c$ is a factor if $f(c)=0$	-
Rational Root Theorem	Possible rational roots	$\pm \frac{p}{q}$

Additional info: This guide covers the main topics listed for Exam 1, including functions, their properties, operations, inverses, complex numbers, quadratic and polynomial equations, and related theorems. Students should review lecture notes, homework, and textbook examples for practice problems and deeper understanding.