Trimester Review Chapter 1 Sections 1.1-1.3

Review for Chapter 1 (Sections 1.1-1.3)

Section 1.1 Functions

 y=f(x)---The point (x,y) is on the graph of f.

A function is a relation that assigns each X-value with exactly one Y-value. The domain is the set of inputs whereas the range is the set of outputs. 

How to test for functions algebraically: 

                                                 Each value of X has exactly one value of Y.

Evaluating Functions:

• 
                                  Given this equation find f(2).    Substitute 2 in for X.

 f(2)= 4+4-1

f(2)=7

 

 A Piecewise-Defined Function:

Evaluate the function when x= -2,1,2

f(-2)=7

Since negative two is less then one I plugged it into the first equation.

  Since both one and two are greater then or equal to one I will plug them into the second equation.

f(1)= 1+4

f(1)=5

f(2)=2+4

f(2)=6

Evaluating a Difference Quotient:

Example: 

*The h denominator always divides out.

Section 1.2 Graphs of Functions

 Finding Domain and Range of a Function:

{(-3,5), (-2,5), (-1,5), (0,5), (1,5), (2,5)} 

Domain: {-3, -2, -1, 0, 1, 2} (listed from smallest to largest)

Range: {5}

Increasing and Decreasing Functions:

A function f is increasing on an interval if for any x1 and x2 in the interval, 

implies 

A function f is decreasing on an interval if for any x1 and x2 in the interval,

implies 

A function f is constant on an interval if for, any x1 and x2 in the interval,

 

Relative Minimum and Maximum Values:

A function value f(a) is called a relative minimum of f if there exists an interval (x1, x2) that contains a such that:

implies

A function value f(a) is called a relative maximum of f if there exists an interval (x1,x2) that contains a such that:

implies 

Even and Odd Functions:

A function f is even, if for each x in the domain of f, f(-x) = f(x).

A function f is odd, if for each x in the domain of f, f(-x) = -f(x).

On the left is an odd function, and on the right is an even function.

Section 1.3: Shifting, Reflecting, and Stretching Graphs

Vertical and Horizontal Shifts:

1. Vertical shift c units upward: h(x)= f(x)+c

2. Vertical shift c units downward: h(x)= f(x) -c

3. Horizontal shift c units to the right: h(x)= f(x-c)

4. Horizontal shift c units to the left: h(x)= f(x+c)

Reflections:

1. Reflection in the x-axis: h(x) = -f(x)

2. Reflection in the y-axis h(x)= f(-x)

Reflection over x-axis.

Reflection across y-axis.