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
)3)(( and ),1)(()),0(()),0(( ggfffggf oo
:8)( and 9)(for Evaluate 3 xxgxxf
))0((gf
))0(( fg
Example 3
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
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.
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
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
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
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
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
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
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
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