Download - Pshs 3rd yr_functions
2. What is the sum h(x) + g(x) if h(x) = 5x – 1 and g(x) = 3x2 + 6x – 7 ?
25 1 3 6 7h x g x x x x23 11 8x x
3. If F(x)= 2x + 3 and G(x) = 5x – 4, what is the composite function
G(F(x))?
5 4G F x F x
5 2 3 4x
10 15 4
10 11
x
x
4. What is the domain of the square root function ?
• DOMAIN
The set of all possible values for x, OR
The x-coordinates of the points of the graph of the function
( )f x x 5
2 ways to analyze:1.) Knowing the graph of the function2.) Using the given equation of the function
The graph of ( )f x x 5
• Courtesy of Wolfram Alpha: www/wolframalpha.com
• GENERAL EQUATION: y a x h k
at x=5
h=horizontal shift
DOMAIN of
is
5x
( )f x x 5
5. Function or not?Relation Function Not a
Functiona.{(5,6), (-2,3), (3,1), (5,2), (8, -4)
}
b. y = 9 – x2
c.
d. x = (y + 6) (y – 3)
-1 1
-2
6. Find the inverse of the function . • HOW TO GET
• 1.) Use y for “f(x)”.
• 2.) Switch x and y.
• 3.) Isolate y.
• 4.) This new y will be your
1f x2 3f x x
1f x
1f x
2 3y x
2 3x y
3
2
xy
13
2
xf x
7. Which of the following is NOT divisible by (x – 1) ?
• 3 Ways to answer:
1.) Using LONG division.
2.) Using SYNTHETIC division.
3.) Using the REMAINDER THEOREM.
Remainder Theorem:
3 2
3 2
3 4
1 3 1 4 0
x x
Therefore,
is divisible by(x - 1).
3 23 4x x
8. What are the roots of the function ?
• ROOTS – also known as the x-coordinates of the x-intercepts, OR the solutions to the equation if h( x ) were equal 0.
• HOW DO WE GET THE x-intercepts?
• Let y = 0, meaning let h( x ) = 0.
3 22 15h x x x x
3 20 2 15x x x
3 2
2
0 2 15
0 2 15
0 5 3
x x x
x x x
x x x
Which is the correct set of roots of f( x )?
x = -3, 0 and 5OR
x = -5, 0 and 3
9. What is the equation of the line that passes through (5, -6) and is perpendicular to the line whose equation is ?• Point (5, -6) is a point on the line we are
looking for.
• Perpendicular means that the slopes of
• and the unknown line are NEGATIVE RECIPROCALS of each other.
4 16 0x y
4 16 0x y
10. Given the function :
• Find the intercepts.
• To get the x-intercept:
• Therefore,
• x-intercept = (1/2, 0).
1 2f x x
0 1 2
2 1
1
2
x
x
x
1 2 0
1
y
y
• To get the y-intercept:
Therefore, y-intercept = (0, 1).
11. Given the function :
• Find the vertex of the parabola.
• 2 ways:
• 1.) Vertex-form of the quadratic equation.
• 2.) Use the vertex formula.
24 1 2f x x
2y a x h k
24: ,
2 4
b ac bvertex
a a
11.b. Find the domain and range.
• Domain – set of all real numbers:
• Range – based on 2 things:
• 1.) the y-coordinate of the VERTEX of the parabola.
• 2.) the direction where the parabola is opening leading coefficient.
• Therefore, the range is 2y
x
Trigonometric functions
• Sine, Cosine, and Tangent
• These are RATIOS of the sides of the right triangle
sinopposite
hypotenuse
cosadjacent
hypotenusetan
opposite
adjacent
SOH CAH TOA
RECALL: Pythagorean Theorem
a
b
c
If side a = 5 cm. and side c = 13 cm., what isthe length of side b?
2 2 2a b c
where a and b are the LEGS, and c is the hypotenuse.
RECALL: The 30-60-90 TRIANGLE THEOREMS
• The side opposite 30 degrees will have a length of ½ of the length of the longest side (hypotenuse).
• The side opposite 60 degrees will be times the length of the longest side.
30
60
3
2
a
b
c
RECALL: THE 45-45-90 TRIANGLE
• In terms of the sides, what kind of triangle is the 45-45-90 triangle?
• The length of the longest side is times the length of a leg.
a
c
a
2
Reference anglesAngle rotation starts from the positive x-axis, then moving counter-clockwise.The reference angle is measured from the x-axis.
What is the reference Angle of the ff?1.) 120 deg =2.) 225 deg =3.) 330 deg =
What IF YOU FORGOT THE CONVERSIONS?
• What is 270 degrees in radian measure?
• Just memorize one thing:
360 270
2 ?
360 2
What are the trigonometric function values of the ff. angles?
Angle Sin x Cos x Tan x Csc x Sec x Cot x
240 deg
-225 deg
-150 deg