Module 1 Review: Properties of Functions

Flashcards reviewing definitions, key features, real-world context models, transformations, and graph properties from Module 1.

What is a one-to-one function?

A function for which each element of the range is paired with exactly one element of the domain.

What is the maximum of a function?

The point that is the highest one on the graph of a function.

What is a linear function?

A function in which no independent variable is raised to a power greater than $1$, written in the form $f(x) = mx + b$, where $m$ and $b$ are real numbers.

What is a continuous function?

A function that can be graphed with a line or an unbroken curve.

What defines odd functions?

Functions that are symmetric in the origin.

What is an identity function?

The function $f(x) = x$.

What is a constant function?

A function of the form $f(x) = a$, where $a$ is any number.

What is an absolute value function?

A function written as $f(x) = |x|$, in which $f(x) ≥ 0$ for all values of $x$.

What is a parent function?

The simplest of the functions in a family that is transformed to create other members in a family of functions.

What is a discrete function?

A discontinuous function in which the domain is a set of individual values, rather than a continuous interval of real numbers.

Salvatore charges $100$ dollars for plumbing work under $2$ hours, $250$ dollars for $2$ to $5$ hours, and $400$ dollars for $5$ to $8$ hours. What best describes the domain of the price scale function?

$$\{x \mid 0 < x < 8\}$$

In Tia's debt repayment table, what are the x-intercept and y-intercept, and what do they mean in context?

The x-intercept is $(6, 0)$, meaning the debt is fully paid off ($0$ dollars owed) at week $6$. The y-intercept is $(0, 80)$, meaning the initial amount borrowed was $80$ dollars.

Why does the end behavior of the trajectory graph of a thrown ball not make sense in a real-world context?

The end behavior does not make sense because the ball cannot have a negative height.

How do the key features of $f(x) = -2x + 4$ and $g(x) = 5x - 5$ compare?

Both functions are linear and continuous.

What absolute value function models the cross-section of the roof with vertex at $(3, 6)$?

$$g(x) = -|x - 3| + 6$$

If $g(x) = f(0.75x)$, how is the graph of $g(x)$ related to the graph of $f(x)$?

The graph of $g(x)$ is a horizontal stretch of the graph of $f(x)$ by a factor of $\frac{3}{4}$.

What transformation occurs from the parent function $f(x) = x^2$ to $j(x) = (x - 4)^2$?

Translation

What piecewise function represents the graph with parts $y = 2x + 1$ for $x \le 2$ and $y = x - 3$ for $x > 2$?

$$g(x) = \begin{cases} 2x + 1, & \text{if } x \le 2 \\ x - 3, & \text{if } x > 2 \end{cases}$$

What is the domain of the function shown in ACT/SAT Practice Question 23?

$$(-\infty, \infty)$$

Which function represents a vertical compression, reflection in the x-axis, and translation down $3$ units in relation to the parent function $f(x) = x^2$?

$$y = -\frac{1}{3}x^2 - 3$$