how do i find the domain and range of equations?
example: f(x)=ab.solute value of x-3.
f(x)=square root 36-x^2
g(x)=-2x+2.
how would i know what the domain and range is of these. could i use a graphing calculator to tell me the domain and range??
y = │x-3│
becomes y = x-3 , y = -x+3
clearly I can use any x I feel like and get a y value.
so the domain is the set of real numbers
A nice way to get the range is to graph both lines and use only the part above the x-axis. These two intersect at (3,0) and form a V upwards
the range is any y ≥ 0
f(x) = √(36-x^2)
clearly if we use any value of x such that -6 < x < 6 we would be taking the square root of a negative, which would be undefined.
so the domain is x ≤ -6 OR x ≥ 6
the range is y ≥ 0
the last one is real easy.
in a simple way, the domain is set of all number you can use for x in your function without causing any undefined results,
the range is the set of resulting y values you get from those x's
so what is the answer to the last one?
To find the domain and range of equations, you can use mathematical methods without necessarily relying on a graphing calculator. Here's how you can determine the domain and range for each of the given equations:
1. f(x) = |x - 3|:
The domain of a function is the set of all possible inputs (values of x) for which the function is defined. In this case, there is no restriction on the values of x and the absolute value function is defined for all real numbers. Therefore, the domain of f(x) is (-∞, +∞).
To find the range, you'll need to consider the behavior of the function. The absolute value function always gives non-negative outputs since it is the distance from zero. So, the range of f(x) = |x - 3| is [0, +∞).
2. f(x) = √(36 - x^2):
For this equation, note that the expression inside the square root (36 - x^2) must be greater than or equal to zero to ensure a real-valued output. Solve this inequality by setting 36 - x^2 ≥ 0:
36 - x^2 ≥ 0
(x - 6)(x + 6) ≥ 0
The critical points are x = -6 and x = 6. Therefore, the domain of f(x) is [-6, 6].
To find the range, observe that the square root of a non-negative number gives only non-negative outputs. Hence, the range of f(x) = √(36 - x^2) is [0, +∞).
3. g(x) = -2x + 2:
This linear equation is defined for all real numbers since there are no restrictions on the values of x. Hence, the domain of g(x) is (-∞, +∞).
Since g(x) represents a straight line with a negative slope, it will produce all possible values of y. Therefore, the range of g(x) = -2x + 2 is also (-∞, +∞).
While a graphing calculator can visually display the domain and range, it is important to understand the underlying principles to calculate them manually.