Graphs of Functions 1.2

Finding the Domain and Range of a Function

Domain: all possible x-coordinates of a graph
Range: all possible y-coordinates of a graph

Finding Domains:

Find the domain of y=x²+x+2

• The domain for this equation would be (-∞,∞) because this equation can be any real number in order to remain a function

Find the domain of y=√(x²-4)

• When you have a square root the number inside the radical can't be negative, otherwise it will result in an imaginary number.


• Pretend like the radical isn't there


• set y=0 so you have x²-4=0


• x=±2


• It's okay for the number under the radical to be zero, but it is not okay for the number under the radical to be negative because that would result in an imaginary number. We know that both 2 and -2 equal zero. By testing out numbers on the number line we find that any number in between -2 and 2 would make the number under the radical negative (not including 2 and -2).


• By this we can tell that all possible domains for this function will be: [-∞,-2) U [2,∞)

Find the domain of the Function: ( x+4)/(x²+3x+2)

• In this Equation for finding the domain the numerator does NOT matter






• Factor the denominator


• x=-2 and -1


• The only restrictions are that the denominator can't be 0.


• The domain of this function is: (-∞,-2) U (-1,∞)

Finding the Range:

Find the Range of the Function: y=x²+2x+1

• We know that this equation is going to be a parabola pointing downward. Therefore this parabola is going to have a minimum range


• By finding the vertex for x, x = -b/2a we will get an answer of -1 to find the minimum value we plug -1 back into the equation and we get 0.


• Since the graph has a minimum the answer for all range values is going to be bigger than 0


• The range for this function is [0,∞)

Increasing and Decreasing Functions

• A function f is increasing on an interval if the slope of the interval is positive.


• A function f is decreasing on an interval if the slope of the interval is negative.


• A function f is constant on an interval if the slope of the graph is a horizontal line.

Even and Odd Functions

• A function is even if f(-x)=f(x). You can tell if a graph is even if it is symmetrical through the y-axis


• A function is odd if f(-x)=-f(x). You can tell if a graph is odd if it is symmetrical through the origin