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Functions and Their Graphs
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Terms in this set (41)
Function
any process that assigns a single value of y to each number of x.
If no vertical line can be drawn so that it intersects more than once, it is a function
Increasing function= going up , left to right.
Decreasing function= going down, left to right
Constant function= horizontal.
Square Root Function
f(x)= square root of x
Domain= (0, inf)
Range= (0, infinity)
Graph: Sideways quadratic
Absolute Value Function
f(x)= x in between two straight lines
Domain= (- infinity, infinity)
Range= (0, infinity)
Graphing a function
1. Find at least 4 ordered pair solutions
2. Plot Points
3. Connect
Cubic Function
f(x)= x cubed
Domain= (- infinity, infinity)
Range= (- infinity, infinity)
Graph= Mirror half Quadratic
Quadratic Function
f(x)= x squared
Domain= (- infinity, infinity)
Range= (0, infinity)
Graph= Parabola or smile that increases quickly
Constant Function
f(x)= C
Domain= (-infinity, infinity)
Range= (yly=0)
Graph= horizontal line
Linear Function
f(x)= x
Domain= (-infinity, infinity)
Range= (-infinity, infinity)
Graph= Diagonal line
Elementary Functions:
Dependent variable
Y
Elementary Functions:
Independent Variable
X
Elementary Functions:
Domain
The set of all the values of x for which the function is defined
Elementary Functions:
Range
The set of the corresponding values of y.
Elementary Functions:
Example:
Evaluate f(1) for y=f(x)= 5x+2
f(x)= 5x+2
f(1)= 5(1) + 2
= 5+2
= 7
Operations on Functions:
Addition, Subtraction, multiplication, and division
a. (f+g) (x)= f(x)+ g(x)
b. (f-g) (x)= f(x)- g(x)
c. (f x g) (x)=f(x) g (x)
d. (f/g) (x)= f(x)/ g(x), g(x)
Operations on Functions
Addition Example f+g:
let f(x)= 2x squared -1 and g(x)= 5x+3
1. f+g
(f+g) (x)
= f(x) + g(x)
= 2x squared -1+ 5x+3
= 2x squared + 5x+2
Operations on Functions
Subtraction Example (f-g):
let f(x)= 2x squared -1 and g(x)= 5x+3
(f-g)(x)= f(x)-g(x)
= 2x squared-1 - (5x+3)
= 2x squared-1-5x-3
= 2x squared- 5x-4
Operations on Functions:
Multiplication Example (f x g):
let f(x)= 2x squared -1 and g(x)= 5x+3
(f x g) (x)
= f(x)g(x)
= ( 2x squared -1)(5x+3)
= 10x cubed + 6x squared -5x-3
Operations on Functions:
Division Example (f/g)
let f(x)= 2x squared -1 and g(x)= 5x+3
(f/g)(x)
= f(x)/g(x)
= (2x squared-1)/ (5x+3)
Composite Function
f to the degree of g is the notation
is defined as (f to the degree of g) (x)= f(g(x))
Ex.
f(x)= 3x and g(x)= 4x+2
Find (f to the degree of g)(x) and (g to the degree of f)(x)
(f to the degree of g) (x)= f(g(x))= 3(4x+2)
= 12x+6
( g to the degree of f) (x)= g(f(x))= 4(3x)+2
12x+2
Inverse Function
1. The inverse of a function, f to the power of -1, is obtained from f by interchanging the x and y(=f(x)) and then solving for y.
2. The functions, f and g are inverses of one another if g ° f= x and f ° g = x.
3. To find g when f is given, interchange x and g in the equation y= f(x) and solve for g (x).
Translation of a function
will move each point of the function a specific number of units left or right, then up or down.
Reflection of a function
the mirror image of the function.
A reflection about the x-axis changes point (x,y) into point (x,-y).
A reflection about the y-axis changes point (x,y) into point (-x,y).
A reflection about the line y=x will move each point P to a new point P' so that the line y= x is the perpendicular bisector of PP with a line above it.
Each point (x,y) becomes (y,x) after the reflection.
Translation of a function:
90 degree rotation of a function
moves each point P to a new point P' above OP' so that OP= OP' and OP with a line above it is perpendicular to a line above OP.
O' stands for Origin, which is (0,0).
Rotation of a Function:
Counterclockwise rotation
each point (x,y) becomes (-y, x).
Rotation of a function
Clockwise rotation
each point (x,y) becomes (y, -x).
Reflection functions example
A function contains a point (-6,3). If this function is reflected about the line y=x, what will the new coordinates be for this point?
(-6,3) becomes (3,-6).
90 degree rotation problem
For point (4,3), what are the new coordinates of (4,3) if the function is rotated ninety degress clockwise?
(4,3) becomes (3,-4).
90 degree rotation function example
Suppose a function contains the point (4,3). What are the new coordinates of this point if the function is rotated ninety degrees counterclockwise.
(4,3) becomes (-3,4).
Translation Function Example:
Given the function f(x)= x to the power of 2 - 5, what is the resulting function g (x) in which each point of f(x) is moved two units to the left and one unit up?
Since each point is moved two units to the left for g(x) we replace each x in f(x) by x+2.
Also, because each points is moved up one unit, we add one to the expression for f (x) to get g (x).
Thus, g (x)= (x+2) to the power of two - 5+1
= (x+2) to the power of 2- y.
Inverse Function Examples:
Find the inverse of
f(x)= 3X+2
1. F(x)= y=3x+2 to find f to the power of -1 (x) interchange x and y.
y= 3y+2
3y=x-2
2. Solve for y
y= (x-2/3)
Inverse Function Examples:
Find the inverse of
f(x)= y=x squared -3
1. f(x)=y=x squared - 3
To find f to the power of -1(x), interchange x and y
x=y squared - 3
y squared = x+3
Solution: y= square root of x+ 3
Drill Questions
If h(x)= (5/x-2), which of the following is equivalent to the inverse of h(x):
A. (x-2/5)
B. (5/x+2)
C. (2x-5/5)
D. (2x+5/x)
D.
The inverse of a function is found by reversing the roles of the two variables, then solving for the new dependent variable.
Ley y- h(x) so that the initial function can be written as
y= (5/x-2).
Interchanging the x and y, we get
x= (5/y-2)
Multiply this equation by y-2 to get (x) (y-2)= 5.
This equation simplifies to xy-2x= 5.
Add 2x to each side to get xy-2x+2x=5+2x.
= xy= 5+ 2x.
Divide both sides by x to get
y= (5+2x/x).
The right side of this equation, which is equivalent to (2x+5/x ) represents the expression for the inverse of h(x).
Drill Question
What is the domain of f(x):
square root of 6-3x
A. x less than or equal to 2
B. x greater than or equal to 2
C. x less than or equal to -2
D. x greater than or equal to -2
A.
The domain of f(x)= squared root of 6-3x is defined as the allowable values f(x) the square root of any function must be at least zero in order to represent a real value.
Domain= solve 6-3x greater than or equal to 0
Subtract 6 from both sides
= -3x/3= -6/-3
= x less than or equal to 2.
Drill Question
A. If f(x)= 2x+3 and g(x)= x cubed - 5, what is the value of f(g(-3))?
A. -61
B. -32
C. 25
D. 47
A.
g(-3)= (-3)cubed -5=-27-5=-32.
Thus f(g(-3))= f(-32)= (2)(-32)+3= -64+3=-61
Composite functions
Example
Find (f to the degree of g)(2) if f (x)= x squared -3 and g(x)= 3x+1
(f to the degree of g)(2)= f(g(2))
g(x)= 3x+1
1. Substitute the value of x with 2 in g(x)
g(2)= 3(2)+1= 7
f(x)= x squared + 3
2. Substitute the value or solution of g(2) in f(x):
f(7) squared-3=49-3=
Solution: 46
The point (-2,-3) is rotated 90 degrees clockwise about the origin, what is the new location?
A. (-3,2)
B. (-3,-2)
C. (3, -2)
D. (3,2).
A.
(y,-x)
x= -2, y= -3
New point is found by interchanging the coordinates, then switching the sign to the new second coordinate.
(-3,2) is the new point
The point (5, -8) is reflected across the line y=x, what is the new location?
A. (-5, 8)
B. -5,-8)
C. (-8, 5)
D. (-8,-5)
C.
the new location of (x,y) is (y,x).
x= 5 y= -8
Current location: (5,-8) turns into (-8,5).
flip the numbers around
The function g(x) is know to have a range of all real numbers except zero. Which of the following expressions could represent g(x)?
A. 4x squared
B. (4/x)
C. (4/ x squared)
D. - squared root of x squared - 4.
B. The range is represented by the g(x) values.
For g(x)= 4/x, g(x) can assume any value (including negative numbers) except zero.
If (4,-1), (5,-3) (6,-9), (---, -16) represents a function. Which one of the following cannot be filled in the blank space?
A. 5
B. 3
C. -4
D. -9
A.
Function as it relates to a set of ordered pairs- any specific first number must be associated with a single second number.
This implies that a set of ordered pairs is not a function if two ordered pairs contain the same first number, but, a second different number.
In order for this set to qualify as a function, we cannot repeat 4,5, or 6 as a first number of the last ordered pair.
For which one of the following functions is f(1)- f(-1)=2?
A. f(x)= x squared- 3x+4
B. f(x)= 4x squared-2
C. f(x)= x squared -2x-7
D. f(x)= 3x squared -y
B.
For choice A, f(1)- 1 squared -3(1) +4=2, but, f(-1) = (-1) squared - 3(-1)+4= 8x.
Choice B is correct because
f(1)= 4 (1) squared -2=2 and f(-1)= 4(-1) squared
2=2
Each point of the f(x)= x squared + 10 is moved 3 units to the left and 2 units down to create a function g(x).
What would be the y-coordinate of a point on the graph of f(x( whose x-coordinate is -1 on the graph of f(x)=
A. 6
B. 7
C. 8
D. 9
D.
1. Substitute -1 for x in the function f(x) to get (-1) squared +1=1+10=11
The corresponding point on the graph of f(x) is (-1,11).
To find the corresponding point for g(x) this point will be moved 3 units to the left and 2 units down.
This point (-1,11) becomes (-4,9) on graph g(x)
y-coordinate= 9
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