SAT Math: Functions Flashcards

Question 1

Define Function Notation f(a).

Accepted Answer

Writing f(a) means substituting the value a for every x in the rule and simplifying, which gives the single output the function pairs with that input.

Question 2

Define f(x) = 0 Versus f(0).

Accepted Answer

Setting f(x) = 0 asks which inputs make the output zero, giving the x-intercepts; f(0) asks for the output at input zero, giving the y-intercept.

Question 3

Define Reading a Function From a Table.

Accepted Answer

Each row of a function table pairs one input with its output, so finding f(k) means locating k in the input column and reading across to its output.

Question 4

Define Reading a Function From a Graph.

Accepted Answer

To find f(k) from a graph, locate x = k on the horizontal axis, move vertically to the curve, then read off the height of that point.

Question 5

Define Domain of a Model in Context.

Accepted Answer

In a word problem the inputs a function may take are limited by reality, such as whole numbers of tickets or times after launch, not just by the algebra.

Question 6

Define Reading the Range From a Graph.

Accepted Answer

Scan the graph from its lowest point to its highest; every y-value the curve actually reaches belongs to the range, and gaps or open endpoints are left out.

Question 7

Define Average Rate of Change.

Accepted Answer

Between inputs a and b, the average rate of change is (f(b) − f(a)) ÷ (b − a): the slope of the straight line joining those two points on the graph.

Question 8

Define Linear Versus Exponential in a Table.

Accepted Answer

When the inputs rise in equal steps, a linear function adds the same amount to each output, while an exponential function multiplies each output by the same factor.

Question 9

Define Exponential Growth Model.

Accepted Answer

A rule of the form f(t) = a(1 + r)^t, where a is the starting amount and r is the percent increase per period written as a decimal.

Question 10

Define Exponential Decay Model.

Accepted Answer

A rule of the form f(t) = a(1 − r)^t that shrinks by the same percentage every period, so its base sits strictly between 0 and 1.

Question 11

Define Growth Factor.

Accepted Answer

The base b in f(t) = a·b^t; every time t rises by one, the output is multiplied by b, so b above 1 signals growth and b below 1 signals decay.

Question 12

Define Exponent for a Doubling Period.

Accepted Answer

When a quantity changes by a fixed factor every k units of time, the exponent becomes t/k so it counts completed periods rather than single time units.

Question 13

Define Vertical Shift f(x) + k.

Accepted Answer

Adding a constant k to every output moves the whole graph k units up when k is positive and k units down when k is negative.

Question 14

Define Horizontal Shift f(x − h).

Accepted Answer

Replacing x with x − h moves the graph h units to the right, and replacing x with x + h moves it h units left, the opposite of what the sign suggests.

Question 15

Define Vertical Stretch a·f(x).

Accepted Answer

Multiplying every output by a constant a with |a| greater than 1 pulls the graph away from the x-axis; a value between 0 and 1 squashes it toward the axis.

Question 16

Define Reflection Over the x-Axis −f(x).

Accepted Answer

Negating every output turns the graph upside down, so each point (x, y) moves to (x, −y) and any x-intercepts stay exactly where they were.

Question 17

Define Reflection Over the y-Axis f(−x).

Accepted Answer

Swapping x for −x mirrors the graph left to right, so each point (x, y) moves to (−x, y) while the y-intercept stays the same.

Question 18

Define Evaluating a Composition f(g(x)).

Accepted Answer

Work from the inside out: evaluate the inner function g at the given input first, then use that result as the input of the outer function f.

Question 19

Define Finding an Inverse Algebraically.

Accepted Answer

Write y = f(x), exchange x and y, then solve the new equation for y; the expression obtained is f⁻¹(x).

Question 20

Define Domain Restriction From Division.

Accepted Answer

A rational function has no value wherever its denominator equals zero, so those input values must be left out of its domain.

Question 21

Define Domain Restriction From a Square Root.

Accepted Answer

Over the real numbers, whatever sits under a square root must be zero or positive, which limits the inputs the function can accept.

Question 22

Define Intersection of Two Graphs.

Accepted Answer

Where the graphs of f and g cross, both functions share the same input and output, so the crossing's x-value solves the equation f(x) = g(x).

Question 23

Define Counting Solutions With a Graph.

Accepted Answer

The number of real solutions of f(x) = c equals the number of times the horizontal line y = c meets the graph of f.

Question 24

Define Evaluating a Piecewise Function.

Accepted Answer

Decide first which interval the input falls in, then use only the rule attached to that interval to work out the output.

Question 25

Define Increasing and Decreasing Intervals.

Accepted Answer

A function rises on an interval where its outputs grow as the inputs move to the right, and falls where its outputs shrink as the inputs move right.

Question 26

Define Maximum Value of a Function.

Accepted Answer

The greatest output a function reaches on its domain or on a stated interval, seen on a graph as the y-coordinate of the highest point.

Question 27

Define Interpreting a Function in Context.

Accepted Answer

In a word problem the input and output carry units, so a statement such as f(3) = 45 has to be translated back into a sentence about the real quantities.

Question 28

Define Constant Function.

Accepted Answer

A rule that returns the same number for every input; its graph is a horizontal line and its rate of change is zero.

Question 29

Define Absolute Value Function.

Accepted Answer

f(x) = |x| returns how far x is from zero, so every output is zero or positive and the graph forms a V with its point at the origin.