3. Functions and Graphs Flashcards
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
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
The notation f(x) describes the values of ____ at _____.
f
x
True or false: y = x² – 1 defines y as a function of x.
True
True or false: y = ± (√x² – 1)
false
If f(x) = x² – 5x + 4, we can find f(x + 6) by replacing each occurence of ______ by ______.
x
x + 6
The graph of a function is the graph of its _______.
ordered pairs
If any vertical line intersects a graph ______, the graph does not define y as a/an ______ of x.
more than once
function
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.
0
0
zeros
True or false: A function can have more than one y-intercept.
false
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)
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
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
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
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
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
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
If f is an odd function, then f(–x) = ______. The graph of an odd function is symmetric with respect to the ______.
–f(x)
origin
A function defined by two or more equations over a specified domain is called a/an _______ function.
piecewise
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
2
–3
0
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)
h
h
True or false: f(x + h) = f(x) + h
false
True or false: f(x + h) = f(x) + f(h)
false
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
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.
2
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.
4
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