Comprehensive Guide to Functions for ACT Math

Function

A relationship between inputs and outputs where every input has exactly one distinct output.

Vertical Line Test

A method to determine if a graph represents a function based on whether a vertical line crosses it at more than one point.

Function Notation

Used to represent functions, written as f(x), which denotes the output of the function for the input x.

Evaluation of Functions

To find the output of a function by substituting a specific value for the input.

Composition of Functions

Combining two functions where the output of one function becomes the input of another.

Domain

The set of all possible input values (x) of a function.

Range

The set of all possible output values (y) of a function.

Restrictions on Domain

Conditions that limit the input values of a function, such as division by zero or taking square roots.

Linear Function

A function that forms a straight line when graphed, generally represented in slope-intercept form.

Slope-Intercept Form

The equation of a linear function expressed as f(x) = mx + b, where m is the slope and b is the y-intercept.

Parallel Lines

Lines that have the same slope and never intersect.

Perpendicular Lines

Lines that have slopes that are negative reciprocals of each other.

Quadratic Functions

Polynomial functions of degree 2, represented in the form f(x) = ax^2 + bx + c.

Vertex of a Parabola

The peak or valley point of a parabola, found using the formula x = -b/(2a).

End Behavior

The behavior of a polynomial function as the input values approach positive or negative infinity.

Radical Functions

Functions that include a root, typically represented as f(x) = √x.

Piecewise Function

A function that applies different rules for different intervals of the independent variable.

Exponential Functions

Functions of the form f(x) = a * b^x, where a is the initial value and b is the growth factor.

Logarithmic Functions

The inverse of exponential functions, expressed as f(x) = log_b(x).

Function Transformations

Changes to the graph of a function resulting from alterations in the function's equation.

Increasing Function

A function where the output values rise as the input values increase.

Decreasing Function

A function where the output values fall as the input values increase.

Intercepts

Points where the graph of a function crosses the axes; includes x-intercepts and y-intercepts.

Asymptotes

Lines that a graph approaches but never touches, commonly seen in rational and exponential functions.

Common Errors in Function Notation

Mistakes such as misinterpreting the effect of operations on functions, like error in domain restrictions.