3. Functions and Graphs Flashcards by David S

Any set of ordered pairs is called a/an ______. The set of all first components of the ordered pair is called the ______. The set of all second components of the ordered pairs is called the _______.

relation

domain

range

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A set of ordered pairs in which each member of the set of first components corresponds to exactly one member of the set of second components is called a/an ______.

function

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The notation f(x) describes the values of ____ at _____.

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True or false: y = x² – 1 defines y as a function of x.

True

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True or false: y = ± (√x² – 1)

false

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If f(x) = x² – 5x + 4, we can find f(x + 6) by replacing each occurence of ______ by ______.

x + 6

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The graph of a function is the graph of its _______.

ordered pairs

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If any vertical line intersects a graph ______, the graph does not define y as a/an ______ of x.

more than once

function

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If the x-intercepts of a function are –1 and 3, then f(–1) = _____ and f(3) = _____. The x-intercepts, –1 and 3, are called the ______ of the function.

zeros

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True or false: A function can have more than one y-intercept.

false

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Assume that f is a function defined on an open interval I and x1 and x2 are any elements in the interval I.

f is increasing on I if f(x1) _______ when x1 < x2.

f is decreasing on I if f(x1) _______ when x1 < x2.

f is constant on I if f(x1) _______.

< f(x2)

> f(x2)

= f(x2)

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If f(a) > f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a relative _______ of f.

maximum

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If f(a) < f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a relative _______ of f.

minimum

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The graph of an equation is symmetric with respect to the _____ if substituting –x for x in the equation results in an equivalent equation.

y-axis

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The graph of an equation is symmetric with respect to the _____ if substituting –y for y in the equation results in an equivalent equation.

x-axis

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The graph of an equation is symmetric with respect to the _____ if substituting –x for x and –y for y in the equation results in an equivalent equation.

origin

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If f is an even function, then f(–x) = _______. The graph of an even function is symmetric with respect to the ______.

f(x)

y-axis

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If f is an odd function, then f(–x) = ______. The graph of an odd function is symmetric with respect to the ______.

–f(x)

origin

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A function defined by two or more equations over a specified domain is called a/an _______ function.

piecewise

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The greatest integer function is defined by int(x) = the greatest integer that is _______.

For example:
int(2.5) =  \_\_\_\_
int(–2.5) = \_\_\_\_
and int(0.5) = \_\_\_\_\_.

less than or equal to x

–3

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The expression

[f(x + h) – f(x)] / h, h ≠ 0,

is called the _______ of the function. We find this expression by replacing x with ______ each time x appears in the function’s equation. Then we subtract ______. After simplifying, we factor _____ from the numerator and divide out identical factors of ______ in the numerator and denominator.

difference quotient

x + h

f(x)

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True or false: f(x + h) = f(x) + h

false

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True or false: f(x + h) = f(x) + f(h)

false

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Data presented in a visual form as a set of points is called a/an ______. A line that best fits this set of points is called a/an ______ line.

scatter plot

regression

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The slope, m, of a line through the distinct points (x1, y1) and (x2, y2) is given by the formula m = ______.

(y2 – y1) / (x2 – x1)

If a line rises from left to right, the line has _____ slope.

positive

If a line falls from left to right, the line has _____ slope.

negative

The slope of a horizontal line is _____.

zero

The slope of a vertical line is ______.

undefined

The point-slope form of the equation of a nonvertical line with slope m that passes through the point (x1, y1) is _____.

y – y1 = m(x – x1)

The slope-intercept of the equation of a line is ______, where m represents the _____ and b represents the ______.

y = mx + b slope y-intercept

In order to graph the line whose equation is y = (2/5)x + 3, begin by plotting the point _____. From this point, we move _____ units up (the rise) and _____ units to the right (the run).

(0, 3) 2 5

The graph of the equation y = 3 is a/an _____ line.

horizontal

The graph of the equation x = –2 is a/an ______ line.

vertical

The equation Ax + By + C = 0, where A and B are not bith zero, is called the ______ form of the equation of a line.

general

If two nonvertical lines are parallel, then they have _____ slope.

the same

If two nonvertical lines are perpendicular, then the product of their slopes is ______.

–1

Consider the line whose equation is y = –(1/3)x + 5. The slope of any line that is parallel to this line is _____. The slope of any line that is perpendicular to this line is _____.

–(1/3) 3

Consider the line whose equation is 2x + y – 6 = 0. The slope of any line that is parallel to this line is ______. The slope of any line that is perpendicular to this line is ______.

–2 1/2

The slope of the line through the distinct points (x1, y1) and (x2, y2) can be interpreted as the rate of change in ______ with respect to ______.

y x

If (x1, f(x1)) and (x2, f(x2)) are distinct points on the graph of a function f, the average rate of change of f from x1 to x2 is ______.

[f(x2) – f(x1)] / (x2 – x1)

The graph of y = f(x) – 5 is obtained by a/an _____ shift of the graph of y = f(x) ______ a distance of 5 units.

vertical down

The graph of y = f(x – 5) is obtained by a/an ______ shift of the graph of y = f(x) _______ a distance of 5 units.

horizontal to the right

The graph of y = –f(x) is the graph of y = f(x) reflected about _______.

x-axis

The graph of y = f(–x) is the graph of y = f(x) reflected about the _______.

y-axis

The graph of y = 5f(x) is obtained by a/an _______ stretch of the graph of y = f(x) by multiplying each of its ______-coordinates by 5.

vertical y

The graph of y = f(1/5x) is obtained by a/an _______ stretch of the graph of y = f(x) by multiplying each of its _______-coordinates by 5.

horizontal x

True or False: The graph of g(x) = √(x + 4) is the graph of f(x) = √x shifted horizontally to the right by 4 units.

false

We exclude from a function's domain real nimbers that cause division by _______.

zero

We exclude from a function's domaine real numbers that result in a square root of a/an ______ number.

negative

(f + g)(x) = ______

f(x) + g(x)

(f – g)(x) = _______

f(x) – g(x)

(fg)(x) = _______

f(x) ∙ g(x)

(f/g)(x) = _______, provided ______ ≠ 0.

f(x) / g(x) g(x)

The domain of f(x) = 5x + 7 consists of all real numbers, represented in interval notation as ______.

(–∞, ∞)

The domain of g(x) = 3 / (x – 2) consists of all real numbers except 2, represented in interval notation as (–∞, 2) ∪ ______

(2, ∞)

The domain of h(x) = 1/x + 7 / (x –3) consists of all real numbers except 0 and 3, represented in interval notation as (–∞, 0) ∪ ______ ∪ ______.

(0, 3) | 3, ∞

The notation f ∘ g, called _____ of the function f with g, is defined by (f ∘ g)(x) = ______

composition f(g(x))

I find (f ∘ g)(x) by replacing each occurence of x in the equation for ____ with _____.

f g(x)

The notation g ∘ f, called the _____ of the function g with f, is defined by (g ∘ f)(x) = _______

composition g(f(x))

I find (g ∘ f)(x) by replacing each occurrence of x in the equation for _____ with ______.

g f(x)

True or false: f ∘ g is the same function as g ∘ f

false

True or false: f(g(x)) = f(x) ∙ g(x)

false

If f(g(x)) = 3 / (g(x) – 4) and g(x) = 8/x, then 0 and ______ must be excluded from the domain of f ∘ g.

The notation f^-1 means the ______ of the function f.

inverse

If the function g is the inverse of the function f, then f(g(x)) = ____ and g(f(x)) = _____.

x x

A function f has an inverse that is a function if there is no _____ line that intersects the graph of f at more than one point. Such a function is called a/an _____ function.

horizontal one-to-one

The graph of f^-1 is a reflection of the graph of f about the line whose equation is ______.

y = x

The distance, d, between the points (x1, y1) and (x2, y2) in the rectangular coordinate system is d = ______.

√[(x2 – x1)² + (y2 – y1)²]

The midpoint of a line segment whose endpoints are (x1, y1) and (x2, y2) is (______, ______).

(x1 + x2) / 2 (y1 + y2) / 2

The set of all points in a plane that are equidistant from a fixed point is a/an ______. The fixed point is called the ______. The distance from this fixed point to any point on the geometric figure is called the ______.

circle center radius

The standard form of the equation of a circle with center (h, k) and radius r is ______

(x – h)² + (y – k)² = r²

The equation x² + y² + Dx + Ey + F = 0 is called the ______ form of the equation of a circle.

general

In the equation (x² + 4x) + (y² – 8y) = 5, we complete the square on x by adding _____ to both sides. We complete the square on y by adding _____ to both sides.

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