Functions and Their Properties

Concept: The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. This is crucial for determining for which x-values a function will yield a valid output.
Key Areas:
- Linear Functions: The domain is all real numbers, represented as $(- ext{∞}, ∞)$ .
- Radical Functions (even index): The expression inside the radical must be greater than 0. This means for a function of the form $f(x) = ext{sqrt}(x)$ , we require x > 0.
- Logarithmic Functions: The argument of the logarithm must be greater than 0. For instance, for $f(x) = ext{log}(x)$ , we have the condition that x > 0.
- Rational Functions: The denominator of the function cannot equal 0, e.g., for $f(x) = rac{1}{x-1}$ , we must have $x <br>eq 1$ .
- Circle Equations: The domain is restricted by the radius and center of the circle described by the equation $x^2 + y^2 = r^2$ where the constraints will define valid x-values based on the y-values.

Concept: Function properties are characteristics that describe a function's behavior and assist in understanding its graph and calculation of outputs.
Key Areas:
- Symmetry:
- Even Functions: Defined by the property $f(x) = f(-x)$ , indicating symmetry about the y-axis.
- Odd Functions: Defined by the property $f(x) = -f(-x)$ , indicating symmetry about the origin.
- Inverse Functions: For two functions $f$ and $g$ , they are inverses if the following conditions are satisfied:
- $f(g(x)) = x$
- $g(f(x)) = x$

Concept: Transformations of functions involve modifying a parent function to create a new graph which can provide deeper insights into functional behavior.
Key Areas:
- Stretches/Compressions:
- Vertical Stretches/Compressions: A vertical transformation can be described by the function $a f(x)$ , where a > 1 stretches the graph vertically and 0 < a < 1 compresses it.
- Horizontal Stretches/Compressions: These can be reflected in functions of form $f(b imes x)$ , where b > 1 compresses the graph horizontally and 0 < b < 1 stretches it horizontally.
- Shifts:
- Vertical Shifts: Reflected in functions of the form $f(x) + c$ , which shifts the graph vertically upwards or downwards based on the sign and value of $c$ .
- Horizontal Shifts: Exhibited in functions of the form $f(x - d)$ which shifts the graph horizontally to the right or left depending on the value of $d$ .
- Writing Equations: This entails applying the transformations mentioned above to derive the new equation of the function from its base form.

Concept: A one-to-one function is defined as a function where each output (y-value) corresponds to exactly one input (x-value). This ensures that the function does not produce the same output for different inputs.
- Key Test: A function fails the Horizontal Line Test if any horizontal line intersects the graph more than once. If a horizontal line intersects the graph once or not at all, then the function is one-to-one.

Concept: This involves simplifying and rewriting algebraic expressions to make them easier to work with or solve.
Key Skills:
- Simplifying Rational Expressions: This includes factoring expressions and canceling common terms to simplify the expression effectively.
- Completing the Square: This technique is used to rewrite quadratic expressions in the form $a(x - h)^2 + k$ , which allows for easier solving or graphing of the equation.

Concept: The aim here is to find the range of values that satisfy a given inequality, which is a key aspect of algebra.
Key Areas:
- Linear Inequalities: To solve linear inequalities, isolate the variable and remember to flip the sign when multiplying or dividing by a negative number.
- Polynomial Inequalities: This involves finding zeros of the polynomial, creating a sign chart, and testing intervals to identify where the inequalities hold true.
- Number-Line Graphs: When graphing solutions on a number line, use open circles for strict inequalities (e.g., >, <) and closed circles for inclusive inequalities (e.g., $<br>eq$ , $<br>eq, <br>eq$ ) to visually represent the solution set.