The expression $\frac{3}{8} + x$ is a linear function. The range of a linear function depends on the domain of the variable $x$.

1. If no restrictions on $x$: If $x$ is any real number (i.e., the domain is $x \in \mathbb{R}$), the range of $\frac{3}{8} + x$ is also all real numbers. This is because as $x$ increases or decreases, $\frac{3}{8} + x$ can take any value from $-\infty$ to $\infty$.

   Hence, the range of $\frac{3}{8} + x$ is: $\text{Range} = \mathbb{R} = (-\infty, \infty)$

2. If there are domain restrictions: If you have specific limits on $x$, the range would depend on those limits.

Would you like any clarification or details on this?

Here are 5 questions to explore related concepts:

1. What is the range of a quadratic function like $x^2 + 2x + 1$?
2. How does the range change for $f(x) = 1/(x+2)$?
3. How can we determine the range of a logarithmic function?
4. How is the range of a piecewise function determined?
5. Can the range of a function be a set of discrete numbers?

Tip: For linear functions without restrictions, the range is always $(-\infty, \infty)$.