`h (x) = (2x+7)/(x-4)`
Which of the following must be true about `h (x)` ?
I. `h (14) =3.5`
II. The domain of h (x) is all real numbers
III. `h(x)` may be positive or negative

Options

A. I and II only
B. I and III only
C. II and III only
D. I, II, and III

Explanation

To solve the problem, we need to analyze the function \( h(x) = \frac{2x + 7}{x - 4} \) and determine the validity of the three statements provided.

### Step 1: Evaluate \( h(14) \)
We start by substituting \( x = 14 \) into the function.

\[ h(14) = \frac{2(14) + 7}{14 - 4} \]

Calculating the numerator and denominator:

\[ = \frac{28 + 7}{10} = \frac{35}{10} = 3.5 \]

**Conclusion for Statement I**: \( h(14) = 3.5 \) is true.

### Step 2: Determine the Domain of \( h(x) \)
The domain of a function is all the values of \( x \) for which the function is defined. The function \( h(x) \) is undefined when the denominator is zero.

Set the denominator equal to zero:

\[ x - 4 = 0 \implies x = 4 \]

Thus, \( h(x) \) is undefined at \( x = 4 \). Therefore, the domain of \( h(x) \) is all real numbers except \( x = 4 \).

**Conclusion for Statement II**: The domain of \( h(x) \) is not all real numbers; it is all real numbers except \( x = 4 \). This statement is false.

### Step 3: Determine if \( h(x) \) can be positive or negative
To check if \( h(x) \) can take both positive and negative values, we can evaluate it at different points.

1. We already found \( h(14) = 3.5 \) (positive).
2. Now, let's evaluate \( h(0) \):

\[ h(0) = \frac{2(0) + 7}{0 - 4} = \frac{7}{-4} = -1.75 \]

This shows that \( h(0) \) is negative.

**Conclusion for Statement III**: \( h(x) \) can be both positive and negative, hence this statement is true.

### Final Conclusion
- Statement I is true.
- Statement II is false.
- Statement III is true.

The statements that must be true about \( h(x) \) are I and III.

### Answer
The correct answer is that statements I and III are true.

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