I Am Getting Confused on How To Find The Domain And Range Of Functions of a Graph.
For Example: f(x)=x^4-8x^2+8.
it has two min. points and one max point.
A function that is a single polynomial will have a domain of all real numbers. For the polynomial function you specified, you must compute the absolute minimum of the function in order to determine the range. The absolute minimum will be the smaller of the two local minima.
for a function of x:
the domain is the set of x-values for with the function is defined.
the range is the set of y-values for which the function is defined.
it is often easier to find the values for which a function is not defined, and say the domain or range is everything but that
with polynomials though, our lives are easy. polynomials are defined for all real x (unless we are restricting the domain with some additional condition).
for an even polynomial where the leading coefficient is positive, the range is $\displaystyle y \in [\text{absolute minimum}, \infty)$
2a. Domain and Range of a Function
In your case the Domain would be all Real numbers.Domain: The complete set of possible values of the independent variable in a function is called the domain of the function.
In plain English, the definition means:Range: The complete set of all possible resulting values of the dependent variable in a function is the range of the function.
The domain of a function is all the possible x values which will make the function "work" by outputting real values.
When finding the domain, remember:
- denominator (bottom) of a fraction cannot be zero
- the values under a square root must be positive
In plain English, the definition means:
The range of a function is the possible y values of a function resulting when we substitute all the possible x-values.
When finding the range, remember:
- substitute different x-values into the expression for y to see what is happening
- make sure you look for minimum and maximum values of y
- draw a sketch!
how are you confused?
no matter what x-values you plug in, the function works. you cover all x-values and so the domain is all real numbers. as for x^3, the function is continuous. one end goes to infinity, the other goes to minus infinity. you cover all y-values. so the domain is all real numbers.
Range can be tricky. Remember that for polynomials with real coefficients and odd degree, the range is all real numbers. For polynomials of even degree, you have to find the absolute minimum if the coefficient of the highest power of x is positive, or the absolute maximum if the coefficient of the highest power of x is negative.
The range is all possible values of y. Since you have said that -8 is the absolute minimum, there can be no values of y under -8 for any value of x.
Take the limit as x goes to positive and negative infinity. They both go to positive infinity. So like you said the range would be -8 to positive infinity.