How do I find the domain and range of the functions?
1. f(x)=2/3x - 4
2. f(x)= x^3-3x+2
3. f(x)= 1/2 l x-2 l
4. f(x)= l x-1 l / x-1
domain of all polynomials is all reals
domain of rational functions is all reals except where the denominator is zero.
1. assuming you meant 2/(3x) - 4
domain: all reals except x=0
range: all reals
2. domain: all reals
range: all reals, since odd-degree
3. domain: all reals
range: y >= 0
4. domain: all reals except x=1
range: {1,-1}
To find the domain and range of a function, you need to understand the definitions of these terms and apply them to each given function. Let's go through each function one by one:
1. f(x) = 2/3x - 4:
- The domain of a function is the set of all possible input values, or x-values, for which the function is defined. In this case, the function is a linear equation, so it is defined for all real numbers. Therefore, the domain is (-∞, ∞).
- The range of a function is the set of all possible output values, or y-values. As the function is a linear equation, it will produce all real numbers, so the range is also (-∞, ∞).
2. f(x) = x^3 - 3x + 2:
- Again, the domain is all real numbers because there are no restrictions on the input values.
- To determine the range, we would need to analyze the behavior of the function. Since this is a cubic function, it will continue indefinitely in both positive and negative directions. Therefore, the range is also (-∞, ∞).
3. f(x) = 1/2 |x - 2|:
- The domain is still all real numbers.
- The absolute value function |x - 2| will always return a non-negative value, so the range of this function will be all non-negative real numbers, including zero. Thus, the range is [0, ∞).
4. f(x) = |x - 1| / (x - 1):
- The domain of this function has one restriction. The denominator cannot be equal to zero, so we need to find when (x - 1) = 0. Solving this equation, we find that x = 1. Therefore, the domain is all real numbers except x = 1.
- The range of this function depends on whether the expression inside the absolute value is positive or negative. If x < 1, the expression inside the absolute value will be negative, and if x > 1, it will be positive. Therefore, the range will be all real numbers except 0, which means (-∞, 0) U (0, ∞).
Remember that understanding the definitions of domain and range is crucial for determining these values for any given function.