Graphs of Fundamental Functions
The following are fundamental functions whose stated
properties and graphs you must
know.
1. The Constant Function
y = f(x) = c
Properties :
(I) Domain: x∈(- ∞, ∞)
(II) Range: y∈{c}or y = c.
(III)
y- intercept : (0, c)
x- intercept : None except for y = f(x) = 0 (In this case the
x-axis is the graph)
(IV) Constant over x∈(- ∞, ∞), that is, always constant
(V) Symmetry: Even (y-axis symmetry)
(VI) End Behavior:
As x → - ∞, y = c
As x → ∞, y = c
(VII) No asymptote. |
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2. The Identity Function
y = f(x) = x
Properties :
(I) Domain: x∈(- ∞, ∞)
(II) Range: y∈(- ∞, ∞).
(III) y- intercept : (0,0); x- intercept : (0,0)
(IV) Increasing over x∈(- ∞, ∞), that is, always increasing
(V) Symmetry: Odd (origin symmetry)
(VI) End Behavior:
As x → - ∞, y → - ∞
As x → ∞, y → ∞
(VII) No asymptote. |
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3. The Absolute Value Function
Properties:
(I) Domain: x∈(- ∞, ∞)
(II) Range: y∈[0, ∞).
(III) y-intercept: (0,0); x-intercept: (0,0)
(IV) Decreasing over x∈(- ∞, 0). Increasing over x∈(0, ∞).
(V) Symmetry: Even (y-axis symmetry)
(VI) End Behavior:
As x → - ∞, y → ∞
As x → ∞, y → ∞
(VII) No asymptote. |
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4. The Square Function
y = f(x) = x2
Properties:
(I) Domain: x∈(- ∞, ∞)
(II) Range: y∈[0, ∞).
(III) y-intercept: (0,0); x-intercept: (0,0)
(IV) Decreasing over x∈(- ∞, 0). Increasing over x∈(0, ∞).
(V) Symmetry: Even (y-axis symmetry)
(VI) End Behavior:
As x → - ∞, y → ∞
As x → ∞, y → ∞
(VII) No asymptote. |
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5.The Cube Function
y = f(x) = x3
Properties:
(I) Domain: x∈(- ∞, ∞)
(II) Range: y∈(- ∞, ∞).
(III) y-intercept: (0,0); x-intercept: (0,0)
(IV) Increasing over x∈(- ∞, ∞); that is, always increasing
(V) Symmetry: Odd (origin symmetry)
(VI) End Behavior:
As x → - ∞, y → - ∞
As x → ∞, y → ∞
(VII) No asymptote. |
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6. The Square - Root Function
Properties:
(I) Domain: x∈[0, ∞)
(II) Range: y∈[0, ∞).
(III) y-intercept: (0,0); x-intercept: (0,0)
(IV) Increasing over x∈(0, ∞).
(V) Symmetry: None
(VI) End Behavior:
As x → 0+, y → 0
As x → ∞, y → ∞
(VII) No asymptote. |
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7. The Reciprocal Function
y = f(x) = 1/x
Properties:
(I) Domain: x∈(- ∞, 0) ∪ (0, ∞). That is, all real numbers except x = 0.
(II) Range: y∈(- ∞, 0) ∪ (0, ∞). That is, all real numbers except y = 0.
(III) y-intercept: None; x-intercept: None
(IV) Decreasing over x∈(- ∞, 0) and over x∈(0, ∞).
(V) Symmetry: Odd (origin symmetry)
(VI) End Behavior:
As x → - ∞, y → 0;
As x →  (approaches 0 from the left), y → - ∞
As x →  (approaches 0 from the right), y → ∞
As x → ∞, y → 0
(VII) Vertical asymptote: x = 0 (y-axis); Horizontal asymptote:
y = 0
(x-axis) |
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8. The Exponential Function
y = f(x) = exProperties:
(I) Domain: x∈(- ∞, ∞)
(II) Range: y∈(0, ∞).
(III) y-intercept: (0,1); x-intercept: None
(IV) Increasing over x∈(- ∞, ∞); that is, always increasing
(V) Symmetry: None
(VI) End Behavior:
As x → - ∞, y → 0
As x → ∞, y → ∞
(VII) Horizontal asymptote: y = 0 (the x-axis). No vertical asymptote. |
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9. The Natural Logarithm Function
y = f(x) = ln(x)
Properties:
(I) Domain: x∈(0, ∞)
(II) Range: y∈(- ∞, ∞).
(III) y-intercept: None; x-intercept: (1,0)
(IV) Increasing over x∈(0, ∞); that is, always increasing
(V) Symmetry: None
(VI) End Behavior:
As x → , y → - ∞
As x → ∞, y → ∞
(VII) Vertical asymptote: x = 0 (the y-axis). No horizontal asymptote. |
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Note:
y = ex and y = ln(x) are inverse functions.
If two functions are inverses of each other then the domain of one is the range
of the
other and vice versa. For example, if (2, -3) is a point on a function, then
(-3, 2) is a point
on its inverse.
To get the graph of the inverse of a function from the graph of the function,
simply reflect
the graph about the line y = x.
So if you start out with y = ex , you can get the graph of y = ln(x), simply
reflect the
graph of y = ex about the line y = x.
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