Function families

Objective

Identify a function's family from its equation or graph and describe its key features.

What it means

A function family is a group of functions that share the same basic shape and parent function, such as linear (f(x) = x), quadratic (f(x) = x^2), cubic (f(x) = x^3), absolute value (f(x) = |x|), square root (f(x) = √x), rational (f(x) = 1/x), and exponential (f(x) = b^x).

Key terms

Parent function

The simplest form of a function family, e.g. f(x) = x^2 for quadratics.

Domain

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

Range

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

Rules to remember

• Linear: f(x) = x, a straight line through the origin with slope 1.
• Quadratic: f(x) = x^2, a U-shaped parabola.
• Cubic: f(x) = x^3, an S-shaped curve.
• Absolute value: f(x) = |x|, a V-shape with vertex at the origin.
• Square root: f(x) = √x, defined only for x ≥ 0.
• Rational: f(x) = 1/x, has a break (asymptote) at x = 0.
• Exponential: f(x) = b^x, grows or decays but never touches the x-axis.

Step by step

1. 1Look at the equation formCheck the highest power, whether there's a root, an absolute value bar, or a variable in the exponent.
2. 2Match to a parent functionCompare the shape to the seven common families.
3. 3Check domain and rangeUse the family's known domain/range to confirm the match.