12 Basic Functions w/ Graphs + Function Rules

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Pre-Calculus

Last updated 9:31 PM on 9/17/22
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20 Terms

1
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The Identity Function

y=x
Domain: ALL REALS
Range: ALL REALS
Discontinuities: None
Decreasing Intervals: None
Increasing Intervals: (-∞,∞)
Symmetry: ODD (about the origin)
Bounded: Unbounded or NONE
Domain: ALL REALS Range: ALL REALS Discontinuities: None Decreasing Intervals: None Increasing Intervals: (-∞,∞) Symmetry: ODD (about the origin) Bounded: Unbounded or NONE
2
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The Squaring Function

y = x^2
Domain: ALL REALS
Range: [0,∞)
Discontinuities: None
Decreasing Intervals: (-∞,0]
Increasing Intervals: [0,∞)
Symmetry: EVEN (across y-axis)
Bounded: Bounded Below
Domain: ALL REALS Range: [0,∞) Discontinuities: None Decreasing Intervals: (-∞,0] Increasing Intervals: [0,∞) Symmetry: EVEN (across y-axis) Bounded: Bounded Below
3
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The Cubing Function

y=x^3
Domain: ALL REALS
Range: ALL REALS
Discontinuities: NONE
Decreasing Intervals: NONE
Increasing Intervals: All REALS
Symmetry: ODD (about the origin)
Bounded: Unbounded or NONE
Domain: ALL REALS Range: ALL REALS Discontinuities: NONE Decreasing Intervals: NONE Increasing Intervals: All REALS Symmetry: ODD (about the origin) Bounded: Unbounded or NONE
4
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The Reciprocal Function
y=1/x
Domain: (-∞,0) ∪ (0,∞)
Range: (-∞,0) ∪ (0,∞)
Discontinuities: NONE **Asymp @ x=0 (VA)**
Decreasing Intervals: (-∞,0)
Increasing Intervals: (0, ∞)
Symmetry: ODD (about the origin)
Bounded: Not Bounded or NONE
Domain: (-∞,0) ∪ (0,∞) Range: (-∞,0) ∪ (0,∞) Discontinuities: NONE **Asymp @ x=0 (VA)** Decreasing Intervals: (-∞,0) Increasing Intervals: (0, ∞) Symmetry: ODD (about the origin) Bounded: Not Bounded or NONE
5
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The Square Root Function

y=√x
Domain: [0,∞)
Range: [0,∞)
Discontinuities: NONE
Decreasing Intervals: NONE
Increasing Intervals: [0, ∞)
Symmetry: NONE
Bounded: Bounded Below
Domain: [0,∞) Range: [0,∞) Discontinuities: NONE Decreasing Intervals: NONE Increasing Intervals: [0, ∞) Symmetry: NONE Bounded: Bounded Below
6
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The Absolute Value Function

y=|x|
Domain: ALL REALS
Range: [0,∞)
Discontinuities: NONE
Decreasing Intervals: (-∞, 0]
Increasing Intervals: [0, ∞)
Symmetry: EVEN (across the y-axis)
Bounded: Bounded Below
Domain: ALL REALS Range: [0,∞) Discontinuities: NONE Decreasing Intervals: (-∞, 0] Increasing Intervals: [0, ∞) Symmetry: EVEN (across the y-axis) Bounded: Bounded Below
7
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The Greatest Integer Function

y=int(x)
Domain: ALL REALS
Range: Integers or "Z"
Discontinuities: Multiple Jumps
Decreasing Intervals: NONE
Increasing Intervals: By Integers
Symmetry: NONE
Bounded: Unbounded or NONE
Domain: ALL REALS Range: Integers or "Z" Discontinuities: Multiple Jumps Decreasing Intervals: NONE Increasing Intervals: By Integers Symmetry: NONE Bounded: Unbounded or NONE
8
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The Exponential Function

y=e^x
Domain: ALL REALS
Range: (0,∞)
Discontinuities: NONE
Decreasing Intervals: NONE
Increasing Intervals: (-∞,∞)
Symmetry: NONE
Bounded: Bounded Below
Domain: ALL REALS Range: (0,∞) Discontinuities: NONE Decreasing Intervals: NONE Increasing Intervals: (-∞,∞) Symmetry: NONE Bounded: Bounded Below
9
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The Natural Log Function

y=ln(x)
Domain: (0,∞)
Range: ALL REALS
Discontinuities: NONE **Asymp @ x=0 (VA)**
Decreasing Intervals: NONE
Increasing Intervals: (-∞,∞)
Symmetry: NONE
Bounded: Unbounded or NONE
Domain: (0,∞) Range: ALL REALS Discontinuities: NONE **Asymp @ x=0 (VA)** Decreasing Intervals: NONE Increasing Intervals: (-∞,∞) Symmetry: NONE Bounded: Unbounded or NONE
10
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The Sine Function

y=sin(x)
Domain: ALL REALS
Range: [-1,1]
Discontinuities: NONE
Decreasing Intervals: It Alternates
Increasing Intervals: It Alternates
Symmetry: ODD (about the origin)
Bounded: Bounded or BOTH @ y=1 & y=-1
Domain: ALL REALS Range: [-1,1] Discontinuities: NONE Decreasing Intervals: It Alternates Increasing Intervals: It Alternates Symmetry: ODD (about the origin) Bounded: Bounded or BOTH @ y=1 & y=-1
11
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The Cosine Function

y=cos(x)
Domain: ALL REALS
Range: [-1,1]
Discontinuities: NONE
Decreasing Intervals: It Alternates
Increasing Intervals: It Alternates
Symmetry: EVEN (across the y-axis)
Bounded: Bounded or BOTH @ y=1 & y=-1
Domain: ALL REALS Range: [-1,1] Discontinuities: NONE Decreasing Intervals: It Alternates Increasing Intervals: It Alternates Symmetry: EVEN (across the y-axis) Bounded: Bounded or BOTH @ y=1 & y=-1
12
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The Logistic Function

y=1/1+e^-x

*feel free to change scale of graph*
Domain: ALL REALS
Range: (0,1)
Discontinuities: NONE **Asymp @ y=0 & y=1 (HA)**
Decreasing Intervals: NONE
Increasing Intervals: ALL REALS
Symmetry: NONE
Bounded: Bounded or BOTH @ y=0 & y=1
Domain: ALL REALS Range: (0,1) Discontinuities: NONE **Asymp @ y=0 & y=1 (HA)** Decreasing Intervals: NONE Increasing Intervals: ALL REALS Symmetry: NONE Bounded: Bounded or BOTH @ y=0 & y=1
13
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y=af(x)
Stretch/Compress graph vertically by scale factor a
14
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y=af(x), |a|>1
Stretch Vertically
15
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y=af(x), |a|<1
Compressed Vertically
16
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y=f(bx)
Compress/Stretch graph horizontally by a factor of 1/b
17
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y=f(bx), |b|>1
Compressed Horizontal
18
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y=f(bx), |b|<1
Stretched Horizontally
19
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- on the outside
Reflect over the x-axis
20
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- on the inside
Reflect over the y-axis