Take simple functions and combine for more

complicated ones Arithmetic - add, subtract, multiply, divide Composition – evaluate one function inside another

15. Combining Functions

Arithmetic Combinations

Given the functions: 2)( and 5)( xxgxxf

Domain:

Domain:

Domain:

Domain:

))(( xgf

))(( xgf

))(( xfg

)(x

g

f

Composition

function.another usingfunction one Evaluating

))(())(( xgfxgf o

Example 1

Find the composition function: ))(( xgf o

15)( and 2 x x g3 135)( xxf

Domain:

Find the composition function: ))(( xgf o

74)( and x x g7

)(

x

xxf

Domain:

Example 2

)3)(( and ),1)(()),0(()),0(( ggfffggf oo

:8)( and 9)(for Evaluate 3 xxgxxf

))0((gf

))0(( fg

Example 3

Use the graphs to evaluate:

)4)(( fg o

)(xg

)(xf

)1)((, gf o

-3 -1

2

-2

4

Example 4

Application

An airplane is flying 300 mi/hr at an altitude of 2 miles. At t = 0, the plane passes directly over a radar station.

Express s as a function of t.

2

d

s

Express s as a function of d.

Express d as a function of t.

One-to-one functions: a function is one-to-one if every input is associated with one output and each output is associated with only one input.

Horizontal Line Test – a function is one-to-one if and only if no horizontal line intersects the graph more than once.

16. Inverse Functions

Inverses

Every one-to-one function, f(x), has an associated Function called an inverse function, f -1(x).

The inverse function reverses what the function does. Its input is another function’s output.Its output is another function’s input.

3

4 0 4

5A B.77

-2

Example 1

25)( xxf

422)8(5)8( f

22)0(5)0( f

?)17(1 f

Finding Inverses Graphically

Inverses swap x and y coordinates.

)(xf

-3 -1

2

-2

4

Finding Inverses Algebraically

Three step process:

The resulting equation is y = f -1(x).

1. Write the equation y = f (x).2. Solve the equation for x in terms of y.3. Swap the x and y variables.

Example 2

Find the inverse function for: 23)( xxf

)(xf

Find the inverse function for: 7)(

x

xxf

Example 3

Inverse Property

:property following thehave and Functions 1-ff

xxff ))((1

xxff ))(( 1

23)( xxf

Graph is a parabola. Either has a minimum or maximum point. That point is called a vertex. Use transformations on x2 and -x2 to get graph of any

quadratic function.

17. Quadratic Functions

Example 1

f(x) = x2: shift up 4 units and shift to the left 5 units

Standard Form

is form standard the)(For 2 cbxaxxf

khxaxf 2)()(

:and vertex, theis ),( where kh

Graph the function: 5)3()( 2 xxf

Minimum value is

Domain:

Range:

Example 2

Graph the function: 3)4()( 2 xxf

Maximum value is

Domain:

Range:

Example 3

Find the maximum or minimum value of the function.

2414)( 2 xxxf

Minimum value is

x-intercepts =

y-intercepts =

Example 4

x value of the vertex

,:form general For the 2 cbxax f(x)

.2

at occurs vertex thea

bx

.2

is min)or (max valueextreme The

a

bf

Find the maximum or minimum value of the functions.

1163)( 2 xxxf

245)( 2 xxxg

Examples

A set of equations involving the same variables A solution is a collection of values that makes

each equation true. Solving a system = finding all solutions

18. Systems of Equations

26

23

435

yx

yx

Is (x, y) = (2, -2) a solution?

Is (1, -1/3) a solution?

Example 1

Substitution Method

Pick one equation and solve for one variable in terms of the other. Substitute that expression for the variable in the other equation. Solve the new equation for the single variable and use that value

to find the value of the remaining variable.

23

435

yx

yx

Example 2

02

2522

yx

yx

Elimination Method Multiply both equations by constants so that one variable has

coefficients that add to zero. Add the equations together to eliminate that variable. Solve the new equation for the single variable and use that value to

find the value of the remaining variable.

23

2053

yx

yx

Example 3

Example 4

252

17432

2

yx

yx

31

A set of linear equations involving the two variables A solution is the intersection of the two lines. One of three things can happen:

19. Systems of Linear Equations

32

Example 1

1224

2736

yx

yx

33

1042

25105

yx

yx

Example 2

34

A chemist wants to mix a 20% saline solution with a 40% saline solution to get 1 liter of a 26% solution. How much of each should she use? (1 liter = 1000 ml)

Example 3

35

A boat travels downstream for 20 miles in 1 hour. It turns around and travels 20 miles upstream (against the current) in 1 hours and 40 minutes. What is the boat’s speed and how fast is the current?

20 miles

Example 4

36

A woman invested in two accounts, one earned 2% and the other earned 10% in simple interest. She put twice as muchin the lower-yielding account. If she earned $3500 in interest last year, how much was invested in each account?

Example 5