find the domain of the following function
g(x)=x/x^2-5x
find the range of the following function
h(x)=x^3+x/x
To find the domain of a function, we need to identify the values of x for which the function is defined. In both cases, we need to look out for any potential division by zero or undefined expressions.
For g(x) = x / (x^2 - 5x), we must consider the denominator (x^2 - 5x). It will be undefined when it equals zero, as division by zero is not allowed. So, let's solve the equation x^2 - 5x = 0 to find the points that could lead to an undefined function.
x^2 - 5x = 0 can be factored as x(x - 5) = 0. Thus, we have two possible values: x = 0 and x = 5. These values would make the denominator equal to zero, causing the function to be undefined.
Therefore, the domain of g(x) is all real numbers except x = 0 and x = 5. In interval notation, the domain can be written as (-∞, 0) U (0, 5) U (5, +∞).
Moving on to h(x) = (x^3 + x) / x, we again need to consider whether any division by zero occurs. In this case, the denominator is x, so the function will be undefined when x = 0.
Hence, the domain of h(x) is all real numbers except x = 0, which can be expressed as (-∞, 0) U (0, +∞).
Now, let's find the range of the function h(x) = (x^3 + x) / x. To determine the range, we must find the possible values for the y-coordinate or output of the function.
Notice that h(x) simplifies to h(x) = x^2 + 1. As x^2 is always non-negative, the smallest value h(x) can take is 1. Therefore, the range of h(x) is all real numbers greater than or equal to 1, denoted as [1, +∞) in interval notation.
In summary:
Domain of g(x) = (-∞, 0) U (0, 5) U (5, +∞)
Domain of h(x) = (-∞, 0) U (0, +∞)
Range of h(x) = [1, +∞)