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Precalculus-Functions-The Basics

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What is a "function?"
A relation in which each x-coordinate is matched with only one y-coordinate.

"y" is a function of "x"
What is the "vertical line test?"
A test that determines if a set of points in a plane represents "y" as a function of "x," which is true if and only if no two points lie on the same vertical line.
What is a "domain?"
The set of the x-coordinates of the points in a function.
What is a "range?"
The set of the y-coordinates of the points in a function.
What does "f(x)" mean? or
g(x)?
or
s(t)?
We read the notation "f(x)" by saying "f of x." This means that your function is dependent on some value of x.

g(x) is read "g of x." The function is dependent on some value of x.

s(t) is read "s of t." The function is dependent on some value of t.
What are the three ways you can determine if a relation is a function?
1. Set of points like: {(-2,1), (1,3), (2,3), (3,-1)}
2. Graph
3. Equation
What kind of notation do you use to describe the domain and range of a function?
Interval Notation

Example 1: (-oo, 4]
Example 2: [-18, 3]
Example 3: (-oo,7)U(7,oo)
To determine if a set of points represents a function, what do you need to do?

Example 1: {(-2,1), (1,3), (2,3), (3,-1)}
Example 2: {(-2,1), (1,3), (1,4), (3,-1)}
You need to check to see if each x-coordinate is matched with ONLY ONE y-coordinate.

Example 1: yes, it is a function
Example 2: no, it is not a function because the points (1,3) and (1,4) show us that the x-coordinate "1" is matched with two y-coordinates, "3" and "4."
To determine if a graph represents a function, what do you need to do?
You need to check to see if the graph passes the Vertical Line Test.

If the vertical line passes through two points on the graph, then the graph does not represent a function.
To determine if an equation represents a function, what do you need to do?
1. Solve for "y" in the equation.
2. Determine whether each choice of "x" will give you ONLY ONE corresponding value of "y."

If you plug in an "x" that can give you two possible answers for "y," then the equation does not represent a function.
What is the expression for the Difference Quotient?
What is the Difference Quotient and what does it tell you?
The Difference Quotient is a measure of the average rate of change (slope) of a function over an interval.
What does something like "f(3)" mean?
f(3) is saying that we need to plug in the value of 3 into a given function.

Example 1: Given f(x)=2x+5 and you are asked to find f(3), this means you plug in "3" for your "x" and solve. So, f(3)=2(3)+5=6+5=11

Example 2: Given f(x)=6x-4x and find f(x+3), then f(x+3)=6(x+3)-4(x+3)=6x+18-4x-12=2x+6
What are "zeros" of a function?
It is an input value (x-coordinate) that gives you an output of zero (y-coordinate).

Example: Given the point (3,0), "3" is the zero of the function.
How do you find the zeros of a polynomial function "f(x)"?
Set the function f(x) = 0 and solve for x.

1. Factor if you need to
2. Solve for x

Example: f(x)=x^2+2x-15
x^2+2x-15=0 set = to 0
(x+5)(x-3)=0 factor to solve
x=-5 and x=3 these are the numbers that if plugged in would make the function = 0.

Zeros are -5 and 3
How do you find the zeros of a rational function "f(x)"?
Set the function f(x) = 0 and solve for x (the same process for a polynomial function).

But, to make your life easier, set ONLY the numerator = 0 and solve for x.

Why? Because any fraction with a numerator of 0 simplifies to 0.

Example 1: 0/7 = 0
Example 2: 0/(x+4) = 0
How do you find the zeros of a radical function "f(x)"?
Set the function f(x) = 0 and solve for x.

1. Isolate the root
2. Square both sides
3. Get x by itself
4. Solve for x

*Make sure to check the solution.
When you look at the graph of a function f(x), how can you tell on which interval it is increasing, decreasing, or constant?
We can relate increasing, decreasing, and constant to more familiar terms like going "uphill," "downhill," and "level."

Increasing = going "uphill"
Decreasing = going "downhill"
Constant = going "level"
What will the graph look like if we say that a function f(x) is:

Increasing on the interval [-00,-1]
Constant on the interval [-1, 1]
Increasing on the interval [1,00]
How do you verify if the function f(x) is odd, even, or neither?
1. The function is odd if f(-x) = -f(x)
2. The function is even if f(-x) = f(x)
3. The function is neither if you don't get either result from 1 or 2 above.
What is a linear function?
A function, that when simplified and you solve for y, you will get the following equation:

y=mx +b OR f(x)=mx+b

m=slope
b=y-intercept
x and y [or f(x)]= coordinates for a point (x,y) OR (x, f(x))on the line
How can you come up with the equation of a linear function f(x) when you are given two points?

Example 1: (3,2) and (5,8)
Example 2: f(3)=2 and f(5)=8
1. Remember a linear equation is y=mx+b
2. m=slope, b=y-intercept, and (x,y) coordinate
3. Solve for slope "m":
m=(y2-y1)/(x2-x1)
4. Solve for "b": substitute one point (x,y) and "m" and solve for b.

Examples 1 and 2:
m=(8-2)/(5-3)=6/2=3

y=mx+b
2=3(3)+b
2=9+b
-7=b

Answer: y=3x-7
What is a piecewise function?
A function f(x) that is given by different expressions on various intervals.

Example: f(x) = x+5 on the interval x<-2
|2x| on the interval -2<x<5
x^2-8 on the interval x>=5
How do you graph a piecewise function?


Example: f(x) = x^2 for the interval (-00,2)
6 for x=2
10-x for the interval (2,6]
Graph each expression and keep only the part of the expression's graph for the specified interval.