Find the domain of the function.
Write your answer using infinity and union.
f(x)=x/x^2+1
f(x)=2x^2/x-3
To find the domain of a function, we need to determine the set of all possible x-values for which the function is defined. In other words, we want to find the values of x that do not result in any issues or contradictions in the function.
Let's start with the function f(x) = x/(x^2 + 1):
The denominator of the function is x^2 + 1, which means that x^2 + 1 ≠ 0, as division by zero is undefined. To solve x^2 + 1 ≠ 0, we subtract 1 from both sides:
x^2 ≠ -1
Since x^2 cannot be negative for real numbers, there are no values that make the denominator equal to zero. Therefore, the entire set of real numbers, from negative infinity to positive infinity, is the domain of the function f(x) = x/(x^2 + 1).
Now let's move on to the function f(x) = 2x^2/(x - 3):
The denominator of the function is x - 3, which means that x - 3 ≠ 0. Solving for x, we add 3 to both sides:
x ≠ 3
Thus, the value of x cannot be equal to 3. However, all other real numbers are valid inputs for this function. Therefore, the domain of f(x) = 2x^2/(x - 3) is all real numbers except x = 3.
Alternatively, we can express the domain using interval notation. For the first function, the domain is (-∞, +∞). For the second function, the domain is (-∞, 3) ∪ (3, +∞). This notation represents the set of all real numbers except x = 3.