FUNCTIONS

TIU College Entrance Test ReviewMath Session 6

1. Evaluate the function for f(-1) if

21 1 1 1f

2( ) 1f x x x

Answers

1 1 11

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 x 23 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 15 410 11

xx

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

is5x

( )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. 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

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

7. 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

8. Given the function :• Find the intercepts.• To get the x-intercept:

• Therefore, • x-intercept = (1/2, 0).

1 2f x x

0 1 22 1

1

2

xx

x

1 2 0

1

y

y

• To get the y-intercept:

Therefore, y-intercept = (0, 1).

Let y = 0, meaning let f( x ) = 0.

let x = 0

8.b. Graph the function.

9. 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 4b ac b

vertexa a

9.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

9.c. Graph the function.

Vertex: (1, 2)

Trigonometric functions

• Sine, Cosine, and Tangent• These are RATIOS of the sides of the right

triangle

sinopposite

hypotenuse

cosadjacent

hypotenuse tan

oppositeadjacent

SOH CAH TOA

RECALL: Pythagorean Theorem

a

b

c

If side a = 5 cm. and side c = 13 cm., what is the 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

32

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

Trigonometric identities

sintan

cos

1

cscsin

1sec

cos

1cot

tan

2 2sin cos 1

THE UNIT CIRCLE

For angle measuresgreater than 90 deg,we use REFERENCE ANGLES.

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 =

UNITS OF ANGLES: DEGREES & RADIANS

360 2 180

902

454

603

306

What IF YOU FORGOT THE CONVERSIONS?

• What is 270 degrees in radian measure?

• Just memorize one thing:

360 2702 ?

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

p. 132 Simplify • The trick is to represent all functions in terms of sin x and cos x. • Recall the definitions of sec x, csc x, and cot x.

sec csc cot .x x x

1sec

cos

1csc

sin

1cot

tan

is only a symbol for the angle being talked about.1

seccos

xx

1csc

sinx

x

1cot

tanx

x

1 1 1sec csc cot

cos sin tanx x x

x x x

1 1 1sec csc cot

cos sin tanx x x

x x x

1 1sintancos

xxx

sin

1cos

xx

cos1

sinxx

cossin

xx

1 1 cossec csc cot

cos sin sinx

x x xx x x

1 1sec csc cot

sin sinx x x

x x

csc cscx x

2csc x2csc x

EXPONENTIAL &LOGARITHMIC FUNCTIONS

Example: Given . What is x?

x = 4.If the bases are the same, then the

exponents will be equal also.

What is x?

43 3x

1 3 26 36x x

3 21 26 6xx

• When the bases are the same, you can equate the exponents already.

3 21 26 6xx

2 3 216 6 xx

1 2 3 2x x 1 6 4x x

4 54

5

x

x

Another example: Solve for x3

2 2 12

64

xx

32 22 64 xx

32 2 42 2xx

32 2 42 2xx

4 32 22 2 xx 2 2 4 3x x

2 4 12 2 6 10

5

3

x xx

x

THE LOGARITHMIC FUNCTION IS THE INVERSE OF THE EXPONENTIAL FUNCTION.

• EXPONENTIAL FORM

• To what exponent will you raise 2 to get 8?

32 ____2log 8 _____8

base

exponent

power of 2

• LOGARITHMIC FORM

?2 8

3

p. 132 examples

31. log 243

102. log 100,000

203. log 20

5

5

1

Example of an exponential function & a logarithmic function

( ) 2xf x

2( ) logf x x