### Instruction: Find the range of the function
\[j(x) = \frac{5x + 3}{x}.\].### Response: I want to find the set of all possible outputs of this function, that is, the values of j(x) for different values of x.
I notice that x cannot be zero, because that would make the denominator zero and the function undefined.
So I exclude x = 0 from the domain of the function.
I also notice that j(x) is a rational function, that is, a ratio of two polynomials.
I recall that the behavior of a rational function depends on the degree of the numerator and denominator polynomials.
In this case, both the numerator and the denominator have degree 1, so the function is a linear function over a linear function.
I also recall that such a function has a horizontal asymptote, which is the ratio of the leading coefficients of the numerator and denominator.
In this case, the leading coefficients are 5 and 1, so the horizontal asymptote is y = 5.
This means that as x gets very large or very small, the function approaches 5, but never equals 5.
So I know that j(x) cannot be 5, but what other values can it be?
I try to find the minimum and maximum values of j(x) on the intervals where it is defined.
On the interval (0, infinity), I see that j(x) is decreasing, because the numerator is increasing slower than the denominator.
So the maximum value of j(x) on this interval is the limit as x approaches 0 from the right, which is infinity.
The minimum value of j(x) on this interval is the limit as x approaches infinity, which is 5.
On the interval (-infinity, 0), I see that j(x) is increasing, because the numerator is decreasing faster than the denominator.
So the minimum value of j(x) on this interval is the limit as x approaches -infinity, which is 5.
The maximum value of j(x) on this interval is the limit as x approaches 0 from the left, which is -infinity.
So I conclude that the range of j(x) is the set of all real numbers except 5, or in interval notation, (-infinity, 5) union (5, infinity).