section 3.4 basic functions constant function linear function identity
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Section 3.4 Basic FunctionsConstant Function ( )f x c
Linear FunctionIdentity
( )f x x
Section 3.4 Basic FunctionsSquare FunctionQuadratic
2( )f x x
Cubic Function 3( )f x x
Section 3.4 Basic Functions
Absolute Value Function
( )f x x
Cube Root Function
3( )f x x
Section 3.4 Basic FunctionsRational Function
1( )f x
x
Square Root Function
2( )f x x
Section 3.4 Basic FunctionsGreatest Integer Function xxxf int
Section 3.4 Basic FunctionsContinuous Functions
Is a function where the graph has no gaps, holes, or breaks…it can be drawn without stopping and lifting your pencil.
Discontinuous Functions
Is a function where the graph has gaps, holes, or breaks…it can not be drawn without stopping and lifting your pencil.
Piecewise-Function
( )f x {for
for
for
Find the following… 2f 3f 1f
Find the inequality that is true for the value of x and plug the value into the function.
1
1
13
x
x
x
1
2
12
2
x
x
21 f 8133 2 f 51222 f
2. What is the domain of the function?Use the individual domain restrictions to find the entire domain. ,3D
Piecewise-Function
( )f x {for
for
for 1
1
13
x
x
x
1
2
12
2
x
x
3. Find all intercepts.Y-intercept is when x = 0 or f(0). 1102 The y-intercept is at ( 0, -1 ).
x-intercepts are when y = f(x) = 0. Take each individual function and set it equal to zero.
012 x 02 012 x12 x
2
1x
Not possible, ½ is not in the domain.
Not true. 12 x1x
x = 1 is ok, but -1 violates domain.
The x-intercept is at ( 1, 0 ).
Piecewise-Function
( )f x {for
for
for 1
1
13
x
x
x
1
2
12
2
x
x
4. Graph the function.Test all endpoints given in the domain restrictions.Color coordinate the functions.
Test the -3 and -1 for the first function.-2(-3) + 1 and -2(-1) + 1 6 + 1 = 7 2 + 1 = 3 ( -3, 7 ) ( -1, 3 )
Closed point because of equal to line.
Open point because of no equal to line.
Test the -1 for the second function. This is a constant function with only x = -1, yields the point ( -1, 2 ).
Test the -1 for the third function, but it will be an open point. This is a quadratic function, so we will test points, 0, 1, 2, etc.
(-1)2 – 1 = 0( -1, 0 ) (0)2 – 1 = 0
( 0, -1 )
(1)2 – 1 = 0( 1, 0 ) (2)2 – 1 = 0
( 2, 3 )
(3)2 – 1 = 0( 3, 8 )
Piecewise-Function
( )f x {for
for
for 1
1
13
x
x
x
1
2
12
2
x
x
5. Use the graph to determine the range.
Use your pencil as a horizontal line. Start at the lowest point on the graph and slide your pencil up. Identify all the y – coordinates that your touching.
1,
6. Is f continuous on its domain? No
Piecewise-Function
( )f x {
1
12
54 2
x
x
x
for
for
for
1
12
2
x
x
x
)2(f
)5(f
)1(f
1. What is f(-2), f(5), and f(-1 )?
15
542 2 52 2 154
121
24
211
2. What is the domain of the function? ,:D
Piecewise-Function
( )f x {
1
12
54 2
x
x
x
for
for
for
1
12
2
x
x
x
4. Graph the function.
1546
4545
5544
4543
1542
54
2
2
2
2
2
2
y
y
y
y
y
xy
215
112
011
1
y
y
y
xy
6
5
4
3
2
x
4121
3120
2121
1122
12
y
y
y
y
xy
1
0
1
2
x
5
2
1
x
Piecewise-Function
( )f x {
1
12
54 2
x
x
x
for
for
for
1
12
2
x
x
x
3. Find all intercepts.
x intercepts
54
54
54
54
054
2
2
2
x
x
x
x
x 0,54
y intercept
3,0 5. Find the Range.
,
6. Is f continuous on its domain? No
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