For $$ h(t)=\frac{t+1}{t-5} $$ (a) Write the domain in set-builder notation. (b) Write the domain in interval notation. Express numbers as integers or simplified fractions. Part: $$ \mathbf{0} / \mathbf{2} $$ Part 1 of 2 (a) Write the domain in set-builder notation. The domain is the empty set. The domain in set-builder notation is $$ \{t \mid t $$ is a real number $$ \} $$ $$ \{t \mid t $$ is a real number and

Question

UpStudy AI · Accepted Answer

To determine the domain of the function $$ h(t) = \frac{t+1}{t-5} $$, we need to identify all real numbers $$ t $$ for which the function is defined. The function is undefined when the denominator is zero because division by zero is not allowed.

### Step-by-Step Reasoning:

1. **Identify the Denominator:**
   $$
   \text{Denominator} = t - 5
   $$

2. **Find When the Denominator is Zero:**
   $$
   t - 5 = 0 \quad \Rightarrow \quad t = 5
   $$
   
   The function is **undefined** at $$ t = 5 $$.

3. **Determine the Domain:**
   - The domain includes all real numbers except $$ t = 5 $$.

### (a) Domain in Set-Builder Notation

Using set-builder notation, the domain can be expressed by specifying all real numbers $$ t $$ such that $$ t $$ is not equal to 5.

$$
\{\, t \mid t \text{ is a real number and } t \neq 5 \,\}
$$

### (b) Domain in Interval Notation

In interval notation, the domain can be represented by two intervals that exclude $$ t = 5 $$:

$$
(-\infty, 5) \cup (5, \infty)
$$

### Final Answer:

**(a) Set-builder notation for the domain:**
$$
\{\, t \mid t \text{ is a real number and } t \neq 5 \,\}
$$
The domain in set-builder notation is $$ \{t \mid t \text{ is a real number and } t \neq 5\} $$.

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Solución

Step-by-Step Reasoning:

(a) Domain in Set-Builder Notation

(b) Domain in Interval Notation

Final Answer:

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