https://file.aiquickdraw.com/mathbot/file/1726100385208de3dun86.jpg

The function given in the image is:

\[
h(x) = \frac{x + 8}{7x}
\]

We are asked to find:
1. **Domain**
2. **Range**
3. **x-intercept**
4. **y-intercept**

### 1. **Domain**:
The function has a denominator, \(7x\), which cannot be zero since division by zero is undefined. Therefore, the domain is all real numbers except where the denominator equals zero, i.e., \(x \neq 0\).

Thus, the domain is:
\[
\text{Domain} = (-\infty, 0) \cup (0, \infty)
\]

### 2. **Range**:
The function can be rewritten as:
\[
h(x) = \frac{x}{7x} + \frac{8}{7x} = \frac{1}{7} + \frac{8}{7x}
\]

As \(x\) approaches infinity, \(h(x)\) approaches \(\frac{1}{7}\). Similarly, as \(x\) approaches zero from either side (positive or negative), \(h(x)\) tends to infinity or negative infinity. Hence, the range covers all real numbers except \(\frac{1}{7}\), since \(h(x)\) cannot exactly reach \(\frac{1}{7}\).

Thus, the range is:
\[
\text{Range} = (-\infty, \frac{1}{7}) \cup (\frac{1}{7}, \infty)
\]

### 3. **x-intercept**:
To find the x-intercept, set \(h(x) = 0\) and solve for \(x\):

\[
\frac{x + 8}{7x} = 0
\]

This equation is zero when the numerator is zero, i.e., \(x + 8 = 0\), which gives \(x = -8\).

Thus, the x-intercept is:
\[
x = -8
\]

### 4. **y-intercept**:
The y-intercept occurs when \(x = 0\). However, since \(x = 0\) is not in the domain of the function (because the denominator would be zero), the function has no y-intercept.

Thus:
\[
\text{No y-intercept}
\]

Would you like further details or clarifications?

Here are 5 related questions to consider:
1. How would the function change if there was a constant added to the denominator?
2. What happens to the range if the function was \(h(x) = \frac{x + 8}{7x - 1}\)?
3. Can this function be plotted easily, and what would its asymptotes look like?
4. What is the behavior of \(h(x)\) as \(x\) approaches positive or negative infinity?
5. How would you determine the inverse of this function?

**Tip:** Always check for vertical asymptotes when dealing with rational functions, as they often indicate domain restrictions.

Given the function h(x) = (x + 8) / (7x), find the domain, range, x-intercept, and y-intercept.

Learn how to analyze the rational function h(x) = (x + 8) / (7x) by finding its domain, range, and intercepts. This problem introduces core algebraic concepts like rational functions and their asymptotes. Suitable for Grades 9-12.

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