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Domain from a graph

Domain is the set of x values that you are allowed to use in the function. From a graph, you are looking for breaks in the function where the graphed function does not cross certain x values. Those are called vertical asymptotes, and you restrict the domain from using those values.

cubic function exponential function
Quadratic/Polynomial: You can use any value you want in a quadratic or polynomial function.In this cubic example, the left side continues towards negative infinity and the right side to positive infinity.Domain: (- ∞, ∞) Exponential: There are no restrictions on exponential domains.Domain: (- ∞, ∞)

Logarithmic:Logarithms are the inverse function of exponential functions. The range values that weren't used for the exponential function aren't defined for the logarithmic function's domain values. Identify the vertical asymptote and the domain will be that value and then the appropriate infinity for the value. In the graph above, the left side has a vertical asymptote at x=-3 and the right side extends to positive infinity. Domain: (-3, ∞) Rationals: From a graph, the vertical asymptotes will identify the values you need to restrict from being used in the domain. There may be some that are holes that you won't be able to see on the graph (need equation for those). When you write it up, say all real numbers except x cannot be equal to and list the vertical asymptotes you see. For the graph above, you can see vertical asymptotes at x = 4 and at x = -3. Domain: all ℜ, x ≠ 4, x ≠ -3.
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Domain from an equation

The two function families that are not going to always be all real numbers are logarithms and rationals. You need to restrict the domain from using x values that aren't defined.
domain logarithms domain rationals
Logarithms are the inverse function for exponential equations. The range of the exponential function (parent) only used positive real numbers, so the domain of its logarithm function (inverse) can't use the negative numbers and 0 in its domain since those weren't defined. If there is a constant added outside the log function, that will be the location of the vertical asymptote with the opposite sign. Write up the domain as from the asymptote to the appropriate infinity. Rationals(factored form) have x values in their denominators. You need to set each factor in the denominator equal to 0, and solve for x. Those will be the x values you can't use. Write it up as all real, x cannot be equal to the values you found when you solved. It doesn't matter if the factor will simplify out. For domain restrictions, just leave it unsimplified and set each factor in the original denominator equal to zero.
Rationals(quadratic equation in denominator)